Characterization of nanoscale structures using an ultrafast electron microscope

ABSTRACT

The present invention relates to methods and systems for 4D ultrafast electron microscopy (UEM)—in situ imaging with ultrafast time resolution in TEM. Single electron imaging is used as a component of the 4D UEM technique to provide high spatial and temporal resolution unavailable using conventional techniques. Other embodiments of the present invention relate to methods and systems for convergent beam UEM, focusing the electron beams onto the specimen to measure structural characteristics in three dimensions as a function of time. Additionally, embodiments provide not only 4D imaging of specimens, but characterization of electron energy, performing time resolved electron energy loss spectroscopy (EELS).

CROSS-REFERENCES TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.13/864,113, filed Apr. 16, 2013, which claims the benefit of U.S. patentapplication Ser. No. 13/565,678, filed Aug. 2, 2012, now U.S. Pat. No.8,440,970, which claims the benefit of U.S. patent application Ser. No.12/575,312, filed Oct. 7, 2009, now U.S. Pat. No. 8,247,769, whichclaims the benefit of U.S. Provisional Patent Application No.61/195,639, filed Oct. 9, 2008, U.S. Provisional Patent Application No.61/236,745, filed Aug. 25, 2009, and U.S. Provisional Patent ApplicationNo. 61/240,946, filed Sep. 9, 2009, which are commonly assigned, thedisclosures of which are hereby incorporated by reference in theirentirety.

U.S. patent application Ser. No. 12/575,285, now U.S. Pat. No. 8,203,120was filed concurrently with U.S. patent application Ser. No. 12/575,312and the entire disclosure of U.S. patent application Ser. No. 12/575,285is hereby incorporated by reference into this application for allpurposes.

STATEMENT AS TO RIGHTS TO INVENTIONS MADE UNDER FEDERALLY SPONSOREDRESEARCH OR DEVELOPMENT

The U.S. Government has certain rights in this invention pursuant toGrant No. GM081520 awarded by the National Institutes of Health, GrantNo. FA9550-07-1-0484 awarded by the Air Force (AFOSR) and Grant No(s).CHE0549936 & DMR0504854 awarded by the National Science Foundation.

BACKGROUND OF THE INVENTION

Electrons, because of their wave-particle duality, can be accelerated tohave picometer wavelength and focused to image in real space. With theimpressive advances made in transmission electron microscopy (TEM),STEM, and aberration-corrected TEM, it is now possible to image withhigh resolution, reaching the sub-Angstrom scale. Together with theprogress made in electron crystallography, tomography, andsingle-particle imaging, today the electron microscope has become acentral tool in many fields, from materials science to biology. For manymicroscopes, the electrons are generated either thermally by heating thecathode or by field emission, and as such the electron beam is made ofrandom electron bursts with no control over the temporal behavior. Inthese microscopes, time resolution of milliseconds or longer, beinglimited by the video rate of the detector, can be achieved, whilemaintaining the high spatial resolution.

Despite the advances made in TEM techniques, there is a need in the artfor improved methods and novel systems for ultrafast electronmicroscopy.

SUMMARY OF THE INVENTION

According to embodiments of the present invention, methods and systemsfor 4D ultrafast electron microscopy (UEM) are provided—in situ imagingwith ultrafast time resolution in TEM. Thus, 4D microscopy providesimaging for the three dimensions of space as well as the dimension oftime. In some embodiments, single electron imaging is introduced as acomponent of the 4D UEM technique. Utilizing one electron packets,resolution issues related to repulsion between electrons (the so-calledspace-charge problem) are addressed, providing resolution unavailableusing conventional techniques. Moreover, other embodiments of thepresent invention provide methods and systems for convergent beam UEM,focusing the electron beams onto the specimen to measure structuralcharacteristics in three dimensions as a function of time. Additionally,embodiments provide not only 4D imaging of specimens, butcharacterization of electron energy, performing time resolved electronenergy loss spectroscopy (EELS).

The potential applications for 4D UEM are demonstrated using examplesincluding gold and graphite, which exhibit very different structural andmorphological changes with time. For gold, following thermally inducedstress, the atomic structural expansion, the nonthermal latticetemperature, and the ultrafast transients of warping/bulging weredetermined. In contrast, in graphite, striking coherent transients ofthe structure were observed in the selected-area image dynamics, andalso in diffraction, directly measuring the resonance period of Young'selastic modulus. Measurement of the Young's elastic modulus for thenano-scale dimension, the frequency is found to be as high as 30gigahertz, hitherto unobserved, with the atomic motions being along thec-axis. Both materials undergo fully reversible dynamical changes,retracing the same evolution after each initiating impulsive stress.Thus, embodiments of the present invention provide methods and systemsfor performing imaging studies of dynamics using UEM.

Other embodiments of the present invention extend four-dimensional (4D)electron imaging to the attosecond time domain. Specifically,embodiments of the present invention are used to generate attosecondelectron pulses and in situ probing with electron diffraction. The freeelectron pulses have a de Broglie wavelength on the order of picometersand a high degree of monochromaticity (ΔE/E₀≈10⁻⁴); attosecond opticalpulses have typically a wavelength of 20 nm and ΔE/E₀≈0.5, where E₀ isthe central energy and ΔE is the energy bandwidth. Diffraction, andtilting of the electron pulses/specimen, permit the direct investigationof electron density changes in molecules and condensed matter. This 4Dimaging on the attosecond time scale is a pump-probe approach in freespace and with free electrons.

As described more fully throughout the present specification, someembodiments of the present invention utilize single electron packets inUEM, referred to as single electron imaging. Conventionally, it wasbelieved that the greater number of electrons per pulse, the better theimage produced by the microscope. In other words, as the signal isincreased, imaging improves. However, the inventor has determined thatby using single electron packets and repeating the imaging process anumber of times, images can be achieved without repulsion betweenelectrons. Unlike photons, electrons are charged and repel each other.Thus, as the number of electrons per pulse increases, the divergence ofthe trajectories increases and resolution decreases. Using singleelectron imaging techniques, atomic scale resolution of motion isprovided once the space-charge problem is addressed.

According to an embodiment of the present invention, a four-dimensionalelectron microscope for imaging a sample is provided. Thefour-dimensional electron microscope includes a stage assemblyconfigured to support the sample, a first laser source capable ofemitting a first optical pulse of less than 1 ps in duration, and asecond laser source capable of emitting a second optical pulse of lessthan 1 ns in duration. The four-dimensional electron microscope alsoincludes a cathode coupled to the first laser source and the secondlaser source. The cathode is capable of emitting a first electron pulseless than 1 ps in duration in response to the first optical pulse and asecond electron pulse of less than 1 ns in response to the secondoptical pulse. The four-dimensional electron microscope further includesan electron lens assembly configured to focus the electron pulse ontothe sample and a detector configured to capture one or more electronspassing through the sample. The detector is configured to provide a datasignal associated with the one or more electrons passing through thesample. The four-dimensional electron microscope additionally includes aprocessor coupled to the detector. The processor is configured toprocess the data signal associated with the one or more electronspassing through the sample to output information associated with animage of the sample. Moreover, the four-dimensional electron microscopeincludes an output device coupled to the processor. The output device isconfigured to output the information associated with the image of thesample.

According to another embodiment of the present invention, a convergentbeam 4D electron microscope is provided. The convergent beam 4D electronmicroscope includes a laser system operable to provide a series ofoptical pulses, a first optical system operable to split the series ofoptical pulses into a first set of optical pulses and a second set ofoptical pulses and a first frequency conversion unit operable tofrequency double the first set of optical pulses. The convergent beam 4Delectron microscope also includes a second optical system operable todirect the frequency doubled first set of optical pulses to impinge on asample and a second frequency conversion unit operable to frequencytriple the second set of optical pulses. The convergent beam 4D electronmicroscope further includes a third optical system operable to directthe frequency tripled second set of optical pulses to impinge on acathode, thereby generating a train of electron packets. Moreover, theconvergent beam 4D electron microscope includes an accelerator operableto accelerate the train of electron packets, a first electron lensoperable to de-magnify the train of electron packets, and a secondelectron lens operable to focus the train of electron packets onto thesample.

According to a specific embodiment of the present invention, a systemfor generating attosecond electron pulses is provided. The systemincludes a first laser source operable to provide a laser pulse and acathode optically coupled to the first laser source and operable toprovide an electron pulse at a velocity v0 directed along an electronpath. The system also includes a second laser source operable to providea first optical wave at a first wavelength. The first optical wavepropagates in a first direction offset from the electron path by a firstangle. The system further includes a third laser source operable toprovide a second optical wave at a second wavelength. The second opticalwave propagates in a second direction offset from the electron path by asecond angle and the interaction between the first optical wave and thesecond optical wave produce a standing wave copropagating with theelectron pulse.

According to another specific embodiment of the present invention, amethod for generating a series of tilted attosecond pulses is provided.The method includes providing a femtosecond electron packet propagatingalong an electron path. The femtosecond electron packet has a packetduration and a direction of propagation. The method also includesproviding an optical standing wave disposed along the electron path. Theoptical standing wave is characterized by a peak to peak wavelengthmeasured in a direction tilted at a predetermined angle with respect tothe direction of propagation. The method further includes generating theseries of tilted attosecond pulses after interaction between thefemtosecond electron packet and the optical standing wave.

According to a particular embodiment of the present invention, a methodof operating an electron energy loss spectroscopy (EELS) system isprovided. The method includes providing a train of optical pulses usinga pulsed laser source, directing the train of optical pulses along anoptical path, frequency doubling a portion of the train of opticalpulses to provide a frequency doubled train of optical pulses, andfrequency tripling a portion of the frequency doubled train of opticalpulses to provide a frequency tripled train of optical pulses. Themethod also includes optically delaying the frequency doubled train ofoptical pulses using a variable delay line, impinging the frequencydoubled train of optical pulses on a sample, impinging the frequencytripled train of optical pulses on a photocathode, and generating atrain of electron pulses along an electron path. The method furtherincludes passing the train of electron pulses through the sample,passing the train of electron pulses through a magnetic lens, anddetecting the train of electron pulses at a camera.

According to an embodiment of the present invention, a method of imaginga sample is provided. The method includes providing a stage assemblyconfigured to support the sample, generating a train of optical pulsesfrom a laser source, and directing the train of optical pulses along anoptical path to impinge on a cathode. The method also includesgenerating a train of electron pulses in response to the train ofoptical pulses impinging on the cathode. Each of the electron pulsesconsists of a single electron. The method further includes directing thetrain of electron pulses along an imaging path to impinge on the sample,detecting a plurality of the electron pulses after passing through thesample, processing the plurality of electron pulses to form an image ofthe sample, and outputting the image of the sample to an output device.

According to another embodiment of the present invention, a method ofcapturing a series of time-framed images of a moving nanoscale object isprovided. The method includes a) initiating motion of the nanoscaleobject using an optical clocking pulse, b) directing an optical triggerpulse to impinge on a cathode, and c) generating an electron pulse. Themethod also includes d) directing the electron pulse to impinge on thesample with a predetermined time delay between the optical clockingpulse and the electron pulse, e) detecting the electron pulse, f)processing the detected electron pulse to form an image, and g)increasing the predetermined time delay between the optical clockingpulse and the electron pulse. The method further includes repeatingsteps a) through g) to capture the series of time-framed images of themoving nanoscale object.

According to a specific embodiment of the present invention, a method ofcharacterizing a sample is provided. The method includes providing alaser wave characterized by an optical wavelength (λ₀) and a directionof propagation and directing the laser wave along an optical path toimpinge on a test surface of the sample. The test surface of the sampleis tilted with respect to the direction of propagation of the laser by afirst angle (α). The method also includes providing a train of electronpulses characterized by a propagation velocity (v_(el)), a spacingbetween pulses

$\left( {\lambda_{0}\frac{v_{e\; 1}}{c}} \right),$and a direction of propagation tilted with respect to the direction ofpropagation of the laser by a second angle (β). The method furtherincludes directing the train of electron pulses along an electron pathto impinge on the test surface of the sample. The first angle, thesecond angle, and the propagation velocity are related by

$\frac{\sin(\alpha)}{\sin\left( {\alpha - \beta} \right)} = {\frac{c}{v_{e\; 1}}.}$

According to another specific embodiment of the present invention, amethod of imaging chemical bonding dynamics is provided. The methodincludes positioning a sample in a reduced atmosphere environment,providing a first train of laser pulses, and directing the first trainof laser pulses along a first optical path to impinge on a sample. Themethod also includes providing a second train of laser pulses, directingthe second train of laser pulses along a second optical path to impingeon a photocathode, and generating a train of electron pulses. One ormore of the electron pulses consist of a single electron. The methodfurther includes accelerating the train of electron pulses andtransmitting a portion of the train of electron pulses through thesample.

Numerous benefits are achieved by way of the present invention overconventional techniques. For example, the present systems providetemporal resolution over a wide range of time scales. Additionally,unlike spectroscopic methods, embodiments of the present invention candetermine a structure in 3-D space. Such capabilities allow for theinvestigation of phase transformation in matter, determination ofelastic and mechanical properties of materials on the nanoscale, and thetime evolution of processes involved in materials and biologicalfunction.

Depending upon the embodiment, one or more of these benefits may beachieved. These and other benefits will be described in more detailthroughout the present specification and more particularly below.

These and other objects and features of the present invention and themanner of obtaining them will become apparent to those skilled in theart, and the invention itself will be best understood by reference tothe following detailed description read in conjunction with theaccompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a simplified diagram of a 4D electron microscope systemaccording to an embodiment of the present invention;

FIG. 2 is a simplified perspective diagram of a 4D electron microscopesystem according to an embodiment of the present invention;

FIG. 2A shows a conceptual design of the system of FIG. 2;

FIG. 2B shows images obtained with ultrafast electron pulses;

FIG. 2C shows high-resolution, phase-contrast images taken with thesystem of FIG. 2;

FIG. 2D shows high-resolution, phase contrast images of chlorinatedcopper pthalocyanine;

FIG. 2E shows images and electron diffraction patterns taken with thesystem of FIG. 2;

FIG. 2F shows energy-filtered UEM images and spectrum;

FIG. 3 is a simplified diagram of a computer system for controlling the4D electron microscope system according to an embodiment of the presentinvention;

FIG. 4 is a simplified block diagram of computer hardware forcontrolling the 4D electron microscope system according to an embodimentof the present invention;

FIG. 5 illustrates time-resolved images, image cross-correlations, anddiffraction for a gold specimen according to an embodiment of thepresent invention;

FIGS. 6A-H illustrate structural dynamics for gold and graphite samples,time-resolved images for a graphite sample, and a plot of intensitydifference for the graphite sample according to an embodiment of thepresent invention;

FIG. 7A illustrates single-electron packets and an electron pulseaccording to an embodiment of the present invention;

FIG. 7B illustrates temporal optical gratings for the generation of freeattosecond electron pulses according to an embodiment of the presentinvention;

FIG. 7C illustrates spatiotemporal compression dynamics of attosecondelectron pulses embodiment of the present invention;

FIG. 8 illustrates space charge effects on pulse width andsynchronization according to an embodiment of the present invention;

FIG. 9 illustrates methods for phase matching laser phase fronts andelectron pulses at extended specimen areas according to an embodiment ofthe present invention;

FIG. 10 illustrates generation of tilted electron pulses according to anembodiment of the present invention;

FIG. 11 illustrates scattering factors of electron energy densitydistributions according to an embodiment of the present invention;

FIG. 12 illustrates schematic diagrams of molecular transitions measuredusing embodiments of the present invention;

FIG. 13 illustrates dispersion of an ultrashort electron packetaccording to an embodiment of the present invention;

FIG. 14 illustrates simplified ray diagrams for spatial and temporallenses according to an embodiment of the present invention;

FIG. 15 illustrates simplified thin lens temporal ray diagrams for thelab and copropagating frames according to an embodiment of the presentinvention;

FIG. 16 a simplified 2D schematic diagram of tiled laser pulsesaccording to an embodiment of the present invention;

FIG. 17 illustrates methods for measuring femtosecond and attosecondcompressed electron packets according to an embodiment of the presentinvention;

FIG. 18 is a simplified schematic diagram of a convergent beam ultrafastelectron microscope according to an embodiment of the present invention;

FIG. 19 illustrates temporal frames measured using the convergent beamultrafast electron microscope illustrated in FIG. 18;

FIG. 20 is a plot of diffraction intensities as a function of time andfluence according to an embodiment of the present invention;

FIG. 21 is a plot of amplitudes of atomic vibrations as a function ofobserved intensity changes at different fluences according to anembodiment of the present invention;

FIG. 22 illustrates images and diffraction patterns of a graphite sampleaccording to an embodiment of the present invention;

FIG. 23 illustrates time-resolved images and difference frames for agraphite sample according to an embodiment of the present invention;

FIG. 24 illustrates time dependence of image cross correlation accordingto an embodiment of the present invention;

FIG. 25 illustrates resonance dynamics and corresponding FFTs of agraphite sample according to an embodiment of the present invention;

FIG. 26 illustrates the structure of phase I Cu(TCNQ), a diffractionpattern obtained using an embodiment of the present invention, and animage of phase I Cu(TCNQ);

FIG. 27 illustrates a tomographic tilt series of images of Cu(TCNQ)single crystals according to an embodiment of the present invention;

FIG. 28 illustrates 4D electron micrographs of a microscale cantileveraccording to an embodiment of the present invention;

FIG. 29 illustrates 4D electron micrographs of a nanoscale cantileveraccording to an embodiment of the present invention;

FIG. 30 illustrates oscillation dynamics and frequencies of amicrocantilever according to an embodiment of the present invention;

FIG. 31 illustrates oscillation dynamics and frequencies of amicrocantilever according to an embodiment of the present invention;

FIG. 32 is a simplified schematic diagram of a ultrafast electronmicroscope-electron energy loss spectroscopy system according to anembodiment of the present invention;

FIG. 33 illustrates a static EEL spectrum according to an embodiment ofthe present invention;

FIG. 34 illustrates the 3D time-energy-amplitude evolution of an EELspectra according to an embodiment of the present invention;

FIG. 35 illustrates the FEELS and UEC temporal behavior according to anembodiment of the present invention;

FIG. 36 illustrates calculated bands of graphite in a portion of theBrillion zone according to an embodiment of the present invention;

FIG. 37 illustrates static and femtosecond-resolved EELS of graphiteaccording to an embodiment of the present invention;

FIG. 38 illustrates EELS peak position and diffraction as a function oftime according to an embodiment of the present invention;

FIG. 39 illustrates a 3D intensity-energy-time FEELS plot according toan embodiment of the present invention; and

FIG. 40 illustrates charge density distributions and crystal structuresaccording to an embodiment of the present invention.

DETAILED DESCRIPTION OF SPECIFIC EMBODIMENTS

Ultrafast imaging, using pulsed photoelectron packets, providesopportunities for studying, in real space, the elementary processes ofstructural and morphological changes. In electron diffraction,ultrashort time resolution is possible but the data is recorded inreciprocal space. With space-charge-limited nanosecond (sub-micron)image resolutions ultrashort processes are not possible to observe. Inorder to achieve the ultrafast resolution in microscopy, the concept ofsingle-electron pulse imaging was realized as a key to the eliminationof the Coulomb repulsion between electrons while maintaining the hightemporal and spatial resolutions. As long as the number of electrons ineach pulse is much below the space-charge limit, the packet can have afew or tens of electrons and the temporal resolution is still determinedby the femtosecond (fs) optical pulse duration and the energyuncertainty, on the order of 100 fs, and the spatial resolution isatomic-scale. However, the goal of full-scale dynamic imaging can beattained only when in the microscope the problems of in situ highspatiotemporal resolution for selected image areas, together with heatdissipation, are overcome.

FIG. 1 is a simplified diagram of a 4D electron microscope systemaccording to an embodiment of the present invention. This diagram ismerely an example, which should not unduly limit the scope of the claimsherein. One of ordinary skill in the art would recognize manyvariations, modifications, and alternatives. As illustrated in FIG. 1, afemtosecond laser 110 or a nanosecond laser 105 is directed through aPockels cell 112, which acts as a controllable shutter. A Glan polarizer114 is used in some embodiments, to select the laser power propagatingin optical path 115. A beam splitter (not shown) is used to provideseveral laser beams to various portions of the system. Although thesystem illustrated in FIG. 1 is described with respect to imagingapplications, this is not generally required by the present invention.One of skill in the art will appreciate that embodiments of the presentinvention provide systems and methods for imaging, diffraction,crystallography, electron spectroscopy, and related fields.Particularly, the experimental results discussed below yield insightinto the varied applications available using embodiments of the presentinvention.

The femtosecond laser 110 is generally capable of generating a train ofoptical pulses with predetermined pulse width. One example of such alaser system is a diode-pumped mode-locked titanium sapphire(Ti:Sapphire) laser oscillator operating at 800 nm and generating 100 fspulses at a repetition rate of 80 MHz and an average power of 1 Watt,resulting in a period between pulses of 12.5 ns. In an embodiment, thespectral bandwidth of the laser pulses is 2.35 nm FWHM. An example ofone such laser is a Mai Tai One Box Femtosecond Ti:Sapphire Laser,available from Spectra-Physics Lasers, of Mountain View, Calif. Inalternative embodiments, other laser sources generating optical pulsesat different wavelengths, with different pulse widths, and at differentrepetition rates are utilized. One of ordinary skill in the art wouldrecognize many variations, modifications, and alternatives.

The nanosecond laser 105 is also generally capable of generating a trainof optical pulses with a predetermined pulse width greater than thatprovided by the femtosecond laser. The use of these two laser systemsenables system miniaturization since the size of the nanosecond laser istypically small in comparison to some other laser systems. By moving oneor more mirrors, either laser beam is selected for use in the system.The ability to select either laser enables scanning over a broad timescale—from femtoseconds all the way to milliseconds. For short timescale measurement, the femtosecond laser is used and the delay stage(described below) is scanned at corresponding small time scales. Formeasurement of phenomena over longer time scales, the nanosecond laseris used and the delay stage is scanned at corresponding longer timescales.

A first portion of the output of the femtosecond laser 110 is coupled toa second harmonic generation (SHG) device 116, for example a bariumborate (BaB₂O₄) crystal, typically referred to as a BBO crystal andavailable from a variety of doubling crystal manufacturers. The SHGdevice frequency doubles the train of optical pulses to generate a trainof 400 nm, 100 fs optical pulses at an 80 MHz repetition rate. SHGdevices generally utilize a nonlinear crystal to frequency double theinput pulse while preserving the pulse width. In some embodiments, theSHG is a frequency tripling device, thereby generating an optical pulseat UV wavelengths. Of course, the desired output wavelength for theoptical pulse will depend on the particular application. The doubledoptical pulse produced by the SHG device propagates along electrongenerating path 118.

A cw diode laser 120 is combined with the frequency doubled opticalpulse using beam splitter 122. The light produce by the cw diode laser,now collinear with the optical pulse produced by the SHG device, servesas an alignment marker beam and is used to track the position of theoptical pulse train in the electron generating path. The collinear laserbeams enter chamber 130 through entrance window 132. In the embodimentillustrated in FIG. 1, the entrance window is fabricated from materialswith high transparency at 400 nm and sufficient thickness to providemechanical rigidity. For example, BK-7 glass about 6 mm thick withanti-reflection coatings, e.g. MgF₂ or sapphire are used in variousembodiments. One of ordinary skill in the art would recognize manyvariations, modifications, and alternatives.

An optical system, partly provided outside chamber 130 and partlyprovided inside chamber 130 is used to direct the frequency doubledoptical pulse train along the electron-generating path 134 inside thechamber 130 so that the optical pulses impinge on cathode 140. Asillustrated, the optical system includes mirror 144, which serves as aturning mirror inside chamber 130. In embodiments of the presentinvention, polished metal mirrors are utilized inside the chamber 130since electron irradiation may damage mirror coatings used on someoptical mirrors. In a specific embodiment, mirror 144 is fabricated froman aluminum substrate that is diamond turned to produce a mirrorsurface. In some embodiments, the aluminum mirror is not coated. Inother embodiments, other metal mirrors, such as a mirror fabricated fromplatinum is used as mirror 144.

In an embodiment, the area of interaction on the cathode was selected tobe a flat 300 μm in diameter. Moreover, in the embodiment illustrated,the frequency doubled optical pulse was shaped to provide a beam with abeam waist of a predetermined diameter at the surface of the cathode. Ina specific embodiment, the beam waist was about 50 μm. In alternativeembodiments, the beam waist ranged from about 30 μm to about 200 μm. Ofcourse, the particular dimensions will depend on the particularapplications. The frequency doubled optical pulse train was steeredinside the chamber using a computer controlled mirror in a specificembodiment.

In a specific embodiment, the optical pulse train is directed toward afront-illuminated photocathode where the irradiation of the cathode bythe laser results in the generation of electron pulses via thephotoelectric effect. Irradiation of a cathode with light having anenergy above the work function of the cathode leads to the ejection ofphotoelectrons. That is, a pulse of electromagnetic energy above thework function of the cathode ejects a pulse of electrons according to apreferred embodiment. Generally, the cathode is maintained at atemperature of 1000 K, well below the thermal emission thresholdtemperature of about 1500 K, but this is not required by the presentinvention. In alternative embodiments, the cathode is maintained at roomtemperature. In some embodiments, the cathode is adapted to provide anelectron pulse of predetermined pulse width. The trajectory of theelectrons after emission follows the lens design of the TEM, namely thecondenser, the objective, and the projector lenses. Depending upon theembodiment, there may also be other configurations.

In the embodiment illustrated, the cathode is a Mini-Vogel mount singlecrystal lanthanum hexaboride (LaB₆) cathode shaped as a truncated conewith a flat of 300 μm at the apex and a cone angle of 90°, availablefrom Applied Physics Technologies, Inc., of McMinnville, Oreg. As isoften known, LaB₆ cathodes are regularly used in transmission andscanning electron microscopes. The quantum efficiency of LaB₆ cathodesis about 10⁻³ and these cathodes are capable of producing electronpulses with temporal pulse widths on the order of 10⁻¹³ seconds. In someembodiments, the brightness of electron pulses produced by the cathodeis on the order of 10⁹ A/cm²/rad² and the energy spread of the electronpulses is on the order of 0.1 eV. In other embodiments, the pulse energyof the laser pulse is reduced to about 500 pJ per pulse, resulting inapproximately one electron/pulse

Generally, the image quality acquired using a TEM is proportional to thenumber of electrons passing through the sample. That is, as the numberof electrons passing through the sample is increased, the image qualityincreases. Some pulsed lasers, such as some Q-switched lasers, reducethe pulse count to produce a smaller number of pulses characterized byhigher peak power per pulse. Thus, some laser amplifiers operate at a 1kHz repetition rate, producing pulses with energies ranging from about 1μJ to about 2 mJ per pulse. However, when such high peak power lasersare used to generate electron pulses using the photoelectric effect,among other issues, both spatial and temporal broadening of the electronpulses adversely impact the pulse width of the electron pulse or packetproduced. In some embodiments of the present invention, the laser isoperated to produce low power pulses at higher repetition rates, forexample, 80 MHz. In this mode of operation, benefits available usinglower power per pulse are provided, as described below. Additionally,because of the high repetition rate, sufficient numbers of electrons areavailable to acquire high quality images.

In some embodiments of the present invention, the laser power ismaintained at a level of less than 500 pJ per pulse to prevent damage tothe photocathode. As a benefit, the robustness of the photoemitter isenhanced. Additionally, laser pulses at these power levels preventspace-charge broadening of the electron pulse width during the flighttime from the cathode to the sample, thus preserving the desiredfemtosecond temporal resolution. Additionally, the low electron countper pulse provided by some embodiments of the present invention reducesthe effects of space charge repulsion in the electron pulse, therebyenhancing the focusing properties of the system. As one of skill in theart will appreciated, a low electron count per pulse, coupled with ahigh repetition rate of up to 80 MHz provided by the femtosecond laser,provides a total dose as high as one electron/Å² as generally utilizedin imaging applications.

In alternative embodiments, other suitable cathodes capable of providinga ultrafast pulse of electrons in response to an ultrafast optical pulseof appropriate wavelength are utilized. In embodiments of the presentinvention, the cathode is selected to provide a work function correlatedwith the wavelength of the optical pulses provided by the SHG device.The wavelength of radiation is related to the energy of the photon bythe familiar relation λ(μm)≈1.24÷ν(eV), where λ is the wavelength inmicrons and ν is the energy in eV. For example, a LaB₆ cathode with awork function of 2.7 eV is matched to optical pulses with a wavelengthof 400 nm (ν=3.1 eV) in an embodiment of the present invention. Asillustrated, the cathode is enclosed in a vacuum chamber 130, forexample, a housing for a transmission electron microscope (TEM). Ingeneral, the vacuum in the chamber 130 is maintained at a level of lessthan 1×10⁻⁶ torr. In alternative embodiments, the vacuum level variesfrom about 1×10⁻⁶ torr to about 1×10⁻¹⁰ torr. The particular vacuumlevel will be a function of the varied applications.

In embodiments of the present invention, the short duration of thephoton pulse leads to ejection of photoelectrons before an appreciableamount of the deposited energy is transferred to the lattice of thecathode. In general, the characteristic time for thermalization of thedeposited energy in metals is below a few picoseconds, thus no heatingof the cathode takes place using embodiments of the present invention.

Electrons produced by the cathode 140 are accelerated past the anode 142and are collimated and focused by electron lens assembly 146 anddirected along electron imaging path 148 toward the sample 150. Theelectron lens assembly generally contains a number of electromagneticlenses, apertures, and other elements as will be appreciated by one ofskill in the art. Electron lens assemblies suitable for embodiments ofthe present invention are often used in TEMs. The electron pulsepropagating along electron imaging path 148 is controlled in embodimentsof the present invention by a controller (not shown, but described inmore detail with reference to certain Figures below) to provide anelectron beam of predetermined dimensions, the electron beam comprisinga train of ultrafast electron pulses.

The relationship between the electron wavelength (λ_(deBroglie)) and theaccelerating voltage (U) in an electron microscope is given by therelationship λ_(deBroglie)=h/(2m₀eU)^(1/2), where h, m₀, e are Planck'sconstant, the electron mass, and an elementary charge. As an example,the de Broglie wavelength of an electron pulse at 120 kV corresponds to0.0335 Å, and can be varied depending on the particular application. Thebandwidth or energy spread of an electron packet is a function of thephotoelectric process and bandwidth of the optical pulse used togenerate the electron packet or pulse.

Electrons passing through the sample or specimen 150 are focused byelectron lens assembly 152 onto a detector 154. Although FIG. 1illustrates two electron lens assemblies 146 and 152, the presentinvention is not limited to this arrangement and can have other lensassemblies or lens assembly configurations. In alternative embodiments,additional electromagnets, apertures, other elements, and the like areutilized to focus the electron beam either prior to or after interactionwith the sample, or both.

Detection of electrons passing through the sample, includingsingle-electron detection, is achieved in one particular embodimentthrough the use of an ultrahigh sensitivity (UHS) phosphor scintillatordetector 154 especially suitable for low-dose applications inconjunction with a digital CCD camera. In a specific embodiment, the CCDcamera was an UltraScan™ 1000 UHS camera, manufactured by Gatan, Inc.,of Pleasanton, Calif. The UltraScan™ 1000 CCD camera is a 4 mega-pixel(2048×2048) camera with a pixel size of 14 μm×14 μm, 16-bitdigitization, and a readout speed of 4 Mpixels/sec. In the embodimentillustrated, the digital CCD camera is mounted under the microscope inan on-axis, below the chamber position. In order to reduce the noise andpicture artifacts, in some embodiments, the CCD camera chip isthermoelectrically cooled using a Peltier cooler to a temperature ofabout −25° C. The images from the CCD camera were obtained withDigitalMicrograph™ software embedded in the Tecnai™ user interface, alsoavailable from Gatan, Inc. Of course, there can be other variations tothe CCD camera, cooler, and computer software, depending upon theembodiment.

FIG. 2 is a simplified perspective diagram of a 4D electron microscopesystem according to an embodiment of the present invention. The systemillustrated in FIG. 2 is also referred to as an ultrafast electronmicroscope (UEM2) and was built at the present assignee. The integrationof two laser systems with a modified electron microscope is illustrated,together with a representative image showing a resolution of 3.4 Åobtained in UEM2 without the field-emission-gun (FEG) arrangement of aconventional TEM. In one embodiment of the system illustrated in FIG. 2,the femtosecond laser system (fs laser system) is used to generate thesingle-electron packets and the nanosecond laser system (ns lasersystem) was used both for single-shot and stroboscopic recordings. Inthe single-electron mode of operation, the coherence volume is welldefined and appropriate for image formation in repetitive events. Thedynamics are fully reversible, retracing the identical evolution aftereach initiating laser pulse; each image is constructed stroboscopically,in seconds, from typically 10⁶ pulses and all time-frames are processedto make a movie. The time separation between pulses can be varied toallow complete heat dissipation in the specimen. Without limitingembodiments of the present invention, it is believed that the electronsin the single electron packets have a transverse coherence length thatis comparable to the size of the object that is being imaged. Since thesubsequent electrons have a coherence length on the order of the size ofthe object, the electrons “see” the whole object at once.

To follow the area-specific changes in the hundreds of images collectedfor each time scan, we obtained selected-area-image dynamics (SAID) andselected-area-diffraction dynamics (SADD); for the former, in realspace, from contrast change and for the latter, in Fourier space, fromchanges of the Bragg peak separations, amplitudes, and widths. It is theadvantage of microscopy that allows us to perform this parallel-imagingdynamics with pixel resolution, when compared with diffraction. As shownbelow, it would not have been possible to observe the selected temporalchanges if the total image were to be averaged over all pixels, in thiscase 4 millions.

As illustrated in FIG. 2, a TEM is modified to provide a train ofelectron pulses used for imaging in addition to the thermionic emissionsource used for imaging of samples. Merely by way of example, an FEITecnai™ G² 12 TWIN, available from FEI Company in Hillsboro, Oreg., maybe modified according to embodiments of the present invention. TheTecnai™ G² 12 TWIN is an all-in-one 120 kV (λ_(deBroglie)=0.0335 Å)high-resolution TEM optimized for 2D and 3D imaging at both room andliquid-nitrogen temperatures. Embodiments of the present inventionleverage capabilities provided by commercial TEMs such as automationsoftware, detectors, data transfer technology, and tomography.

In particular, in some embodiments of the present invention, afive-axis, motor-driven, precision goniometer is used with computersoftware to provide automated specimen tilt combined with automatedacquisition of images as part of a computerized tomography (CT) imagingsystem. In these embodiments, a series of 2D images are captured atvarious specimen positions and combined using computer software togenerate a reconstructed 3D image of the specimen. In some embodiments,the CT software is integrated with other TEM software and in otherembodiments, the CT software is provided off-line. One of ordinary skillin the art would recognize many variations, modifications, andalternatives.

In certain embodiments in which low-electron content electron pulses areused to image the sample, the radiation damage is limited to the transitof the electrons in the electron pulses through the sample. Typically,samples are on the order of 100 nm thick, although other thicknesseswould work as long as certain electrons may traverse through the sample.Thus, the impact of radiation damage on these low-electron contentelectron pulse images is limited to the damage occurring during thistransit time. Radiation induced structural damage occurring on longertime scales than the transit time will not impact the collected image,as these damage events will occur after the structural information iscollected.

Utilizing the apparatus described thus far, embodiments of the presentinvention provide systems and methods for imaging material andbiological specimens both spatially and temporally with atomic-scalespatial resolution on the order of 1 nm and temporal resolution on theorder of 100 fs. At these time scales, energy randomization is limitedand the atoms are nearly frozen in place, thus methods according to thepresent invention open the door to time-resolved studies of structuraldynamics at the atomic scale in both space and time. Details of thepresent computer system according to an embodiment of the presentinvention may be explained according to the description below.

Referring to FIG. 2, a photograph of a UEM2 in accordance withembodiments of the present invention is illustrated, together with ahigh-resolution image of graphitized carbon. As illustrated, two lasersystems (fs and ns) are utilized to provide a wide range of temporalscales used in 4D electron imaging. A 200-kV TEM is provided with atleast two ports for optical access to the microscope housing. Using oneor more mirrors (e.g., two mirrors), it is possible to switch betweenthe laser systems to cover both the fs and ns experiments. The opticalpulses are directed to the photocathode to generate electron packets, aswell as to the specimen to initiate (clock) the change in images with awell-defined delay time Δt. The time axis is defined by variable delaybetween the electron generating and clocking pulses using the delaystage 170 illustrated in FIG. 1.

Details of development of ultrafast electron microscopy withatomic-scale real-, energy-, and Fourier-space resolutions is nowprovided. The second generation UEM2 described in FIG. 2 providesimages, diffraction patterns, and electron-energy spectra, and hasapplication for nanostructured materials and organometallic crystals.The separation between atoms in direct images, and the Braggspots/Debye-Scherrer rings in diffraction, are clearly resolved, and theelectronic structure and elemental energies in the electron-energy-lossspectra (EELS) and energy-filtered-transmission-electron microscopy(EFTEM) are obtained.

The development of 4D ultrafast electron microscopy and diffraction havemade possible the study of structural dynamics with atomic-scale spatialresolution, so far in diffraction, and ultrashort time resolution. Thescope of applications is wide-ranging with studies spanning diffractionof isolated structures in reactions (gas phase), nanostructures ofsurfaces and interfaces (crystallography), and imaging of biologicalcells and materials undergoing first-order phase transitions. Typically,for microscopy the electron was accelerated to 120 keV and fordiffraction to 30 keV, respectively, and issues of group velocitymismatch, in situ clocking (time zero) of the change, and framereferencing were addressed. One powerful concept implemented is that of“tilted pulses,” which allow for the optimum resolution to be reached atthe specimen.

For ultrafast electron microscopy, the concept of “single-electron”imaging is fundamental to some embodiments. The electron packets, whichhave a well-defined picometer-scale de Broglie wave length, aregenerated in the microscope by femtosecond optical pulses (photoelectriceffect) and synchronized with other optical pulses to initiate thechange in a temperature jump or electronic excitation. Because thenumber of electrons in each packet is one or a few, the Coulombrepulsion (space charge) between electrons is reduced or eliminated andthe temporal resolution can reach the ultimate, that of the opticalpulse. The excess energy above the work function determines the electronenergy spread and this, in principle, can be minimized by tuning thepulse energy. The spatial resolution is then only dependent on the totalnumber of electrons because for each packet the electron “interfereswith itself” and a coherent buildup of the image is achievable.

The coherence volume, given by:V _(c)=λ_(deBroglie) ²(R/a)²ν_(e)(h/ΔE)establishes that the degeneracy factor is much less than one and thateach Fermionic electron is independent, without the need of thestatistics commonly used for Bosonic photons. The volume is determinedby the values of longitudinal and transverse coherences; V_(c) is on theorder of 10⁶ nm³ for typical values of R (distance to the source), a(source dimension), v_(e) (electron velocity), and ΔE (energy spread).Unlike the situation in transmission electron microscopy (TEM),coherence and image resolution in UEM are thus determined by propertiesof the optical field, the ability to focus electrons on the ultrashorttime scale, and the operational current density. For both “singleelectron” and “single pulse” modes of UEM, these are importantconsiderations for achieving the ultimate spatio-temporal resolutionsfor studies of materials and biological systems.

Atomic-scale resolution in real-space imaging can be achieved utilizingthe second generation ultrafast electron microscopy system (UEM2) ofFIG. 2. With UEM2, which operates at 200 keV (λ_(de Broglie)=2.507 pm),energy-space (electron-energy-loss spectroscopy, EELS) and Fourier-space(diffraction) patterns of nanostructured materials are possible. Theapparatus can operate in the scanning transmission electron microscope(STEM) mode, and is designed to explore the vast parameter spacebridging the gap between the two ideal operating modes ofsingle-electron and single-pulse imaging. With these features, UEM2studies provide new limits of resolution, image mapping, and elementalanalysis. Here, demonstrated are the potential by studying goldparticles and islands, boron nitride crystallites, and organometallicphthalocyanine crystals.

FIG. 2A displays the conceptual design of UEM2, which, as with the firstgeneration (UEM1—described generally in FIG. 1), comprises a femtosecondlaser system and an electron microscope modified for pulsed operationwith femtosecond electron packets. A schematic representation ofoptical, electric, and magnetic components are shown. The optical pulsetrain generated from the laser, in this case having a variable pulsewidth of 200 fs to 10 ps and a variable repetition rate of 200 kHz to 25MHz, is divided into two parts, after harmonic generation, and guidedtoward the entries of the design hybrid electron microscope. Thefrequency-tripled optical pulses are converted to the correspondingprobe electron pulses at the photocathode in the hybrid FEG, whereas theother optical pump beam excites (T-jump or electronic excitation) in thespecimen with a well-defined time delay with respect to the probeelectron beam. The probe electron beam through the specimen can berecorded as an image (normal or filtered, EFTEM), a diffraction pattern,or an EEL spectrum. The STEM bright-field detector is retractable whenit is not in use.

The laser in an embodiment is a diode-pumped Yb-doped fiberoscillator/amplifier (Clark-MXR; in development), which producesultrashort pulses of up to 10 μJ at 1030 nm with variable pulse width(200 fs-10 ps) and repetition rate (200 kHz-25 MHz). The output pulsespass through two successive nonlinear crystals to be frequency doubled(515 nm) and tripled (343 nm). The harmonics are separated from theresidual infrared radiation (IR) beam by dichroic mirrors, and thefrequency-tripled pulses are introduced to the photocathode of themicroscope for generating the electron pulse train. The residual IRfundamental and frequency-doubled beams remain available to heat orexcite samples and clock the time through a computer-controlled opticaldelay line for time-resolved applications.

The electron microscope column is that of a designed hybrid 200-kV TEM(Tecnai 20, FEI) integrated with two ports for optical access, oneleading to the photocathode and the other to the specimen. The fieldemission gun (FEG) in the electron-generation assembly adapts alanthanum hexaboride (LaB₆) filament as the cathode, terminating in aconical electron source truncated to leave a flat tip area with adiameter of 16 μm. The tip is located in a field environment controlledby suppressor and extractor electrodes. The gun can be operated aseither a thermal emission or a photoemission source.

The optical pulses are guided to the photocathode as well as to thespecimen by a computer-controlled, fine-steering mirror in anexternally-mounted and x-ray-shielded periscope assembly. Each laserbeam can be focused to a spot size of <30 μm full width at half maximum(FWHM) at its respective target when the beam is expanded to utilize theavailable acceptance angle of the optical path. Various pulse-energy,pulse-length, and focusing regimes have been used in the measurementsreported here. For UEM measurements, the cathode was heated to a levelbelow that needed to produce detectable thermal emission, as detailedbelow, and images were obtained using both the TEM and the UEM2 mode ofoperation.

For applications involving EELS andenergy-filtered-transmission-electron microscopy (EFTEM), the GatanImaging Filter (GIF) Tridiem, of the so-called post-column type, wasattached below the camera chamber. The GIF accepts electrons passingthrough an entrance aperture in the center of the projection chamber.The electron beam passes through a 90° sector magnet as shown in FIG.2A, which bends the primary beam through a 10 cm bending radius andthereby separates the electrons according to their energy into an energyspectrum. An energy resolution of 0.87 eV was measured for the EELSzero-loss peak in thermal mode operation of the TEM. A retractable slitis located after the magnet followed by a series of lenses. The lensesrestore the image or diffraction pattern at the entrance aperture andfinally it can be recorded on a charge-coupled device (CCD) camera(UltraScan 1000 FT) at the end of the GIF with the Digital Micrographsoftware. The digital camera uses a 2,048×2,048 pixel CCD chip with 14μm square pixels. Readout of the CCD is done as four independentquadrants via four separate digitizing signal chains. This 4-portreadout camera combines single-electron sensitivity and 16-bit pixeldepth with high-speed sensor readout (4 Mpix/s).

Additionally, for scanning-transmission-electron microscopy (STEM), theUEM2 is equipped with a bright-field (BF) detector with a diameter of 7mm and an annular dark-field (ADF) detector with an inner diameter of 7mm and an outer diameter of 20 mm. Both detectors are located in thenear-axis position underneath the projection chamber. The BF detectorusually collects the same signal as the TEM BF image, i.e., thetransmitted electrons, while the ADF detector collects an annulus athigher angle where only scattered electrons are detected. The STEMimages are recorded with the Tecnai Imaging & Analysis (TIA) software.

To observe the diffraction pattern, i.e., the back focal plane of theobjective lens, we inserted a selected area aperture into the imageplane of the objective lens, thus creating a virtual aperture in theplane of the specimen. The result is a selected area diffraction (SAD)pattern of the region of interest only. Adjustment of the intermediateand projector lens determines the camera length. Diffraction patternsare processed and analyzed for crystal structure determination.

Several features of the UEM2 system are worthy of note. First, the highrepetition rate amplified laser source allows us to illuminate thecathode with 343 nm pulses of energies above 500 nJ, compared withtypical values of 3 nJ near 380 nm for UEM1. Thus, a level of averageoptical power for electron generation comparable to that of UEM1operating at 80 MHz, but at much lower repetition rates, was able to bedelivered. The pulse energy available in the visible and IR beams isalso at least two orders of magnitude greater than for UEM1, allowingfor exploration of a much greater range in the choice of sampleexcitation conditions.

Second, the hybrid 200-kV FEG, incorporating an extractor/suppressorassembly providing an extractor potential of up to 4 kV, allows higherresolving power and greater flexibility and control of the conditions ofelectron generation. Third, with simple variation of optical pulsewidth, the temporal and spatial resolution can be controlled, dependingon the requirements of each experiment. Fourth, with variation ofspacing between optical pulses without loss of pulse energy, a widerange of samples can be explored allowing them to fully relax theirenergy after each excitation pulse and rewind the clock precisely; withenough electrons, below the space-charge limit, single-pulse recordingis possible. Finally, by the integration of the EELS spectrometer, thesystem is empowered with energy resolution in addition to the ultrafasttime resolution and atomic-scale space resolution.

The following results demonstrate the capabilities of UEM2 in threeareas: real-space imaging, diffraction, and electron energy resolution.Applications of the present invention are not limited to theseparticular examples. First discussed are the images recorded in the UEMmode, of gold particles and gold islands on carbon films. FIGS. 2Ba-fare UEM2 images obtained with ultrafast electron pulses. Shown are goldparticles (a, d) and gold islands (c, f) on carbon films. UEM2background images (b, e) obtained by blocking thephotoelectron-extracting femtosecond laser pulses. For the UEM2 imagesof gold particles, we used the objective (contrast) aperture of 40 μm toeliminate diffracted beams, while no objective aperture was used for thegold-island images.

FIGS. 2Ba and 2Bd show gold particles of uniform size dispersed on acarbon film. From the higher magnification image of FIG. 2Bd,corresponding to the area indicated by the black arrow in FIG. 2Ba, itis found that the gold particles have a size of 15 nm, and the minimumparticle separation seen in the image is 3 nm.

It should be noted that FIGS. 2Bb and 2Be were recorded under identicalconditions to FIGS. 2Ba and 2Bd, respectively, but without cathodeirradiation by the femtosecond laser pulses. No images were observed,demonstrating that non-optically generated electrons from our warmcathode were negligible. Similar background images with the light pulsesblocked were routinely recorded and checked for all cathode conditionsused in this study.

The waffle (cross line) spacing of the cross grating replica (goldislands) seen in FIG. 2Bc is known to be 463 nm. The gold islands areobserved in FIG. 2Bf, where the bright regions correspond to theamorphous carbon support film and the dark regions to thenanocrystalline gold islands. It is found that the islands may beinterconnected or isolated, depending on the volume fraction of thenanocrystalline phases.

To test the high-resolution capability of UEM utilizing phase contrastimaging, an organometallic compound, chlorinated copper phthalocyanine(hexadecachlorophthalocyanine, C₃₂Cl₁₆CuN₈), was investigated. The majorspacings of lattice fringes of copper of this molecule in projectionalong the c-axis are known to be 0.88, 1.30, and 1.46 nm, with atomicspacings of 1.57 and 1.76 nm.

FIGS. 2Ca-b are high-resolution, phase-contrast UEM images. Shown are animage in FIG. 2Ca and digital diffractogram in FIG. 2Cb of anorganometallic crystal of chlorinated copper phthalocyanine. Thediffractogram was obtained by the Fourier transform of the image in FIG.2Ca. The high-resolution image was taken near the Scherzer focus foroptimum contrast, which was calculated to be 90.36 nm for a sphericalaberration coefficient Cs of the objective lens of 2.26 mm. Theobjective aperture was not used.

FIG. 2Da exhibits the lattice fringes observed by UEM, where the blacklines correspond to copper layers parallel to the c-axis. The Fouriertransform of FIG. 2Da is shown in FIG. 2Db, discussed below, and theclear reciprocity (without satellite peaks in the F.T.) indicates thehigh degree of order in crystal structure.

FIG. 2D shows high-resolution, phase-contrast UEM image and structure ofchlorinated copper phthalocyanine. The high-resolution image shown inFIG. 2Da is a magnified view of the outlined area in FIG. 2Ca. Therepresentation of the crystal structure shown in FIG. 2Db is shown inprojection along the c axis, and the assignment of the copper planesobserved in FIG. 2Da is indicated by the gray lines. The spheres are thecopper atoms.

FIG. 2Da is an enlargement of the area outlined in FIG. 2Ca, clearlyshowing the lattice fringe spacing of 1.46 nm, corresponding to thecopper planes highlighted in gray in FIG. 2Db, in which a unit cell isshown in projection along the c-axis. Regions without lattice fringesare considered to correspond to crystals with unfavorable orientation,or amorphous phases of phthalocyanine, or the carbon substrate. It isknown that in high resolution images, the lattice fringes produced bythe interference of two waves passing through the back focal plane,i.e., the transmitted and diffracted beams, are observed only incrystals where the lattice spacing is larger than the resolution of theTEM. In the profile inset of FIG. 2Da, it should be noted that the FWHMwas measured to be approximately 7 Å, directly indicating that our UEMhas the capability of sub-nanometer resolution.

The digital diffractogram obtained by the Fourier transform of theobserved high-resolution image of FIG. 2Ca is shown in FIG. 2Cb. In thedigital diffractogram, the peaks represent the fundamental spatialfrequency of the copper layers (0.69 nm⁻¹), and higher harmonicsthereof. A more powerful means of obtaining reciprocal-space informationsuch as this is the direct recording of electron diffraction, alsoavailable in UEM.

FIGS. 2Ea-f show measured and calculated electron diffraction patternsof gold islands and boron nitride (BN) on carbon films, along with thecorresponding real-space images of each specimen, all recorded by UEM.Shown are images and measured and calculated electron diffractionpatterns of gold islands (a,b,c) and boron nitride (BN) (d,e,f) oncarbon films. The incident electron beam is parallel to the [001]direction of the BN. All diffraction patterns were obtained by using theselected-area diffraction (SAD) aperture, which selected an area 6 μm indiameter on the specimen. Representative diffraction spots were indexedas indicated by the arrowheads.

In FIG. 2Eb, the electron diffraction patterns exhibit Debye-Scherrerrings formed by numerous diffraction spots from a large number offace-centered gold nanocrystals with random orientations. The rings canbe indexed as indicated by the white arrowheads. The diffraction patternof BN in FIG. 2Ee is indexed by the hexagonal structure projected alongthe [001] axis as shown in FIG. 2Ef. It can be seen that there areseveral BN crystals with different crystal orientations, besides thatresponsible for the main diffraction spots indicated by the whitearrowheads.

In order to explore the energy resolution of UEM, we investigated the BNspecimen in detail by EELS and EFTEM. FIG. 2F shows energy-filtered UEMimages and spectrum. FIG. 2F shows a zero-loss filtered image (FIG.2Fa), boron K-edge mapping image (FIG. 2Fb), thickness mapping image(FIG. 2Fc), and corresponding electron-energy-loss (EEL) spectrum (FIG.2Fd) of the boron nitride (BN) sample. The 5.0- and 1.0-mm entranceaperture were used for mapping images and EEL spectrum, respectively.The thickness at the point indicated by the asterisk in FIG. 2Fc isestimated to be 41 nm. ZL stands for zero-loss.

The boron map was obtained by the so-called three-window method. In theboron map of FIG. 2Fb, image intensity is directly related to arealdensity of boron. In the thickness map of FIG. 2Fc, the brightnessincreases with increasing thickness: d (thickness)=λ(β)ln(I_(t)/I₀),where λ is the mean free path for inelastic scattering under a givencollection angle β, I₀ is the zero-loss (ZL) peak intensity, and I_(t)is the total intensity. The thickness in the region indicated by theasterisk in FIG. 2Fc was estimated to be 41 nm. In the EEL spectrum ofFIG. 2Fd, the boron K-edge, carbon K-edge, and nitrogen K-edge areobserved at the energy of 188, 284, and 401 eV, respectively. In theboron K-edge spectrum, sharp π* and σ* peaks are visible. The carbonK-edge spectrum is considered to result from the amorphous carbon filmdue to the existence of small and broad peaks at the position π* and Γ*,being quite different from spectra of diamond and graphite.

With the capabilities of the UEM2 system described herein, structuraldynamics can be studied, as with UEM1, but with the new energy andspatial resolution are achieved here. Specimens will be excited in aT-jump or electronic excitation by the femtosecond laser pulses (FIG.2A) scanned in time with respect to the electron packets which willprobe the changes induced in material properties through diffraction,imaging, or electron energy loss in different regions, including that ofCompton scattering. Also planned to be explored is the STEM feature inUEM, particularly the annular dark-field imaging, in which compositionalchanges are evident in the contrast (Z contrast). Such images are knownto offer advantages over high-resolution TEM (relative insensitivity tofocusing errors and ease of interpretation). Electron fluxes will beoptimized either through changes of the impinging pulse fluence or bydesigning new photocathode materials. In this regard, with higherbrightness the sub-angstrom limit should be able to be reached. Thepotential for applications in materials and biological research is rich.

FIG. 3 is a simplified diagram of a computer system 310 that is used tooversee the system of FIGS. 1 and 2 according to an embodiment of thepresent invention. This diagram is merely an example, which should notunduly limit the scope of the claims herein. One of ordinary skill inthe art would recognize many other modifications, alternatives, andvariations. As shown, the computer system 310 includes display device320, display screen 330, cabinet 340, keyboard 350, and mouse 370. Mouse370 and keyboard 350 are representative “user input devices.” Mouse 370includes buttons 380 for selection of buttons on a graphical userinterface device. Other examples of user input devices are a touchscreen, light pen, track ball, data glove, microphone, and so forth.

The system is merely representative of but one type of system forembodying the present invention. It will be readily apparent to one ofordinary skill in the art that many system types and configurations aresuitable for use in conjunction with the present invention. In apreferred embodiment, computer system 310 includes a Pentium™ classbased computer, running Windows™ NT, XP, or Vista operating system byMicrosoft Corporation. However, the system is easily adapted to otheroperating systems such as any open source system and architectures bythose of ordinary skill in the art without departing from the scope ofthe present invention. As noted, mouse 370 can have one or more buttonssuch as buttons 380. Cabinet 340 houses familiar computer componentssuch as disk drives, a processor, storage device, etc. Storage devicesinclude, but are not limited to, disk drives, magnetic tape, solid-statememory, bubble memory, etc. Cabinet 340 can include additional hardwaresuch as input/output (I/O) interface cards for connecting computersystem 310 to external devices external storage, other computers oradditional peripherals, which are further described below.

FIG. 4 is a more detailed diagram of hardware elements in the computersystem of FIG. 3 according to an embodiment of the present invention.This diagram is merely an example, which should not unduly limit thescope of the claims herein. One of ordinary skill in the art wouldrecognize many other modifications, alternatives, and variations. Asshown, basic subsystems are included in computer system 310. In specificembodiments, the subsystems are interconnected via a system bus 375.Additional subsystems such as a printer 374, keyboard 378, fixed disk379, monitor 376, which is coupled to display adapter 382, and othersare shown. Peripherals and input/output (I/O) devices, which couple toI/O controller 371, can be connected to the computer system by anynumber of means known in the art, such as serial port 377. For example,serial port 377 can be used to connect the computer system to a modem381, which in turn connects to a wide area network such as the Internet,a mouse input device, or a scanner. The interconnection via system busallows central processor 373 to communicate with each subsystem and tocontrol the execution of instructions from system memory 372 or thefixed disk 379, as well as the exchange of information betweensubsystems. Other arrangements of subsystems and interconnections arereadily achievable by those of ordinary skill in the art. System memory,and the fixed disk are examples of tangible media for storage ofcomputer programs, other types of tangible media include floppy disks,removable hard disks, optical storage media such as CD-ROMS and barcodes, and semiconductor memories such as flash memory,read-only-memories (ROM), and battery backed memory.

Although the above has been illustrated in terms of specific hardwarefeatures, it would be recognized that many variations, alternatives, andmodifications can exist. For example, any of the hardware features canbe further combined, or even separated. The features can also beimplemented, in part, through software or a combination of hardware andsoftware. The hardware and software can be further integrated or lessintegrated depending upon the application. Further details of thefunctionality, which may be carried out using a combination of hardwareand/or software elements, of the present invention can be outlined belowaccording to the figures.

Embodiments of the present invention enable ultrafast imaging withapplications in studies of structural and morphological changes insingle-crystal gold and graphite films, which exhibit entirely differentdynamics, as discussed below. For both, the changes were initiated by insitu femtosecond impulsive heating, while image frames and diffractionpatterns were recorded in the microscope at well-defined times followingthe temperature-jump. The time axis in the microscope is independent ofthe response time of the detector, and it is established using avariable delay-line arrangement; a 1-μm change in optical path of theinitiating (clocking) pulse corresponds to a time step of 3.3 fs.

FIG. 5 illustrates both time-resolved images and diffraction. In thisexample, the images in FIGS. 5A and 5B were obtained stroboscopically atseveral time delays after heating with the fs pulse (fluence of 1.7mJ/cm²). The specimen is a gold single crystal film mounted on astandard 3-mm 400-mesh grid. Shown are the bend contours (dark bands),{111} twins (sharp straight white lines) and holes in the sample (brightwhite circles). The insets in FIG. 5B are image-difference framesIm(t_(ref); t) with respect to the image taken at −84 ps. The goldthickness was determined to be 8 nm by electron energy loss spectroscopy(EELS).

FIG. 5C illustrates the time dependence of image cross-correlations ofthe full image from four independent scans taken with different timesteps. A fit to biexponential rise of the 1 ps step scan is drawn,yielding time constants of 90 ps and 1 ns. FIG. 5D illustrates the timedependence of image cross-correlations at 1 ps time steps for the fullimage and for selected regions of interest SAI #1, #2, and #3, as shownin FIG. 5A. FIGS. 5E and 5F are diffraction patterns obtained using asingle pulse of 6×10⁶ electrons at high peak fluence (40 mJ/cm²) andselected-area aperture of 25 μm diameter. Two frames are given toindicate the change. Diffraction spots were indexed and representativeindices are shown as discussed below.

FIGS. 5A and 5B illustrate representative time-framed images of the goldnanocrystal using the fs excitation pulses at a repetition rate of 200kHz and peak excitation fluence of ˜1.7 mJ/cm². In FIG. 5A, taken at −84ps, before the clocking pulse (t=0), typical characteristic features ofthe single crystal gold in the image are observed: twins and bendcontours. Bend contours, which appear as broad fuzzy dark lines in theimage, are diffraction contrast effects occurring in warped or buckledsamples of constant thickness. In the dark regions, the zone axis (thecrystal [100]) is well aligned with the incident electron beam andelectrons are scattered efficiently, whereas in the lighter regions thealignment of the zone axis deviates more and the scattering efficiencyis lower. Because bend contours generally move when deformation causestilting of the local crystal lattice, they provide in images a sensitivevisual indicator of the occurrence of such deformations.

At positive times, following t=0, visual dynamical changes are observedin the bend contours with time steps from 0.5 ps to 50 ps. A series ofsuch image frames with equal time steps provide a movie of themorphological dynamics. To more clearly display the temporal evolution,image-difference frames were constructed. Depicted as insets in theimages of FIG. 5B, are those obtained when referencing to the −84 psframe; for t=+66 ps and +151 ps. In the difference images, the regionsof white or black directly indicate locations of surface morphologychange (bend contour movement), while gray regions are areas where thecontrast is unchanged from that of the reference frame. It is noted thatthe white and black features in the difference images are nm-scaledynamical change, indicating the size of the induced deformations. Carewas taken to insure the absence of long-term specimen drifts as they cancause apparent contrast change.

To quantify the changes in the image the following method ofcross-correlation was used. The normalized cross correlation of an imageat time t with respect to that at time t′ is expressed as:

${\gamma(t)} = \frac{\sum\limits_{x,y}^{\;}{{C_{x,y}(t)}{C_{x,y}\left( t^{\prime} \right)}}}{\sqrt{\sum\limits_{x,y}^{\;}{{C_{x,y}(t)}^{2}{\sum\limits_{x,y}^{\;}{C_{x,y}\left( t^{\prime} \right)}^{2}}}}}$where the contrast C_(x,y)(t)=[I_(x,y)(t)−Ī(t)]/Ī(t); I_(x,y)(t) andI_(x,y)(t′) are the intensities of pixels at the position of (x,y) attimes t and t′, and Ī(t) and Ī(t′) are the means of I_(x,y)(t) andI_(x,y)(t′), respectively. This correlation coefficient γ(t) is ameasure of the temporal change in “relief pattern” between the twoimages being compared, which can be used as a guide to image dynamics asa function of time. Two types of cross-correlation plots were made,those referenced to a fixed image frame before t=0 and others that showcorrelation between adjacent time points. (Another quantity that showstime dependence qualitatively similar to that of the imagecross-correlation is the standard deviation of pixel intensity indifference images).

FIGS. 5C and 5D show the cross-correlation values between the image ateach measured time point and a reference image recorded before thearrival of the clocking pulse. The experiments were repeated, fordifferent time-delay steps (500 fs, 1 ps, 5 ps, and 50 ps), and similarresults were obtained, showing that morphology changes are completelyreversible and reproducible over each 5 μs inter-pulse interval. Theadjacent-time cross-correlations reveal the timescales for intrinsicchanges in the images, which disappear for time steps below 5 ps,consistent with full-image rise in time. Over all pixels, the time scalefor image change covers the full range of time delay, from ps to ns,indicating the collective averaging over sites of the specimen; as shownin FIG. 5C the overall response can be fit to two time constants of 90ps and 1 ns.

The power of selected area image dynamics (SAID) is illustrated when thedynamics of the bend contours are followed in different selected areasof the image, noted in the micrographs as SAI #1, 2, and 3. Thecorresponding image cross-correlations (FIG. 5D) have different shapeand amplitude from each other and from the full image correlation. Thelarge differences observed here and for other data sets, includingonsets delayed in time and sign reversals, indicate the variation inlocal deformation dynamics. In FIGS. 5G-L, a time-resolved SAI at highermagnification is depicted. A broad and black “penguin-like” contour isobserved as the dominant feature of this area. As shown in the frames, acolossal response to the fs heating is noted. The gray region inside theblack contour appears and broadens with time. Also, a new black contourabove the large central white hole begins to be evident at 1200 ps, andgains substantial intensity over the following 50 ps. All frames takencan be used to construct a movie of SAID.

The observed SAID changes correspond to diffraction contrast(bright-field) effects in bend contours, as mentioned above. It is knownthat the shape of bend contours can be easily altered by sample tiltingor heating inside the microscope. However, here in the ultrafastelectron microscope (UEM) measurements, the changes in local tilt aretransient in nature, reflecting the temporal changes of morphology andstructure. Indeed, when the experiments were repeated in the TEM mode ofoperation, i.e., for the same heating laser pulse and same scanning timebut with continuous electron probe beam, no image change was observed.This is further supported by the change in diffraction observed at highfluences and shown in FIGS. 5E and 5F for two frames, at negative timeand at +50 ns; in the latter, additional Bragg spots are visible, adirect evidence of the transient structural change due to bulging atlonger times.

Whereas real-space imaging shows the time-dependent morphology, theselected area diffraction dynamics (SADD) patterns provide structuralchanges on the ultrashort timescale. Because the surface normal of thefilm is parallel to the [100] zone axis, the diffraction pattern of thesample was properly indexed by the face-centered-cubic (fcc) structureprojected along the [100] zone axis at zero tilt angle (see FIG. 5E).From the positions of the spots in FIG. 5F, which are reflections fromthe {113} and {133} planes, forbidden in the [100] zone-axis viewing, wemeasured the interplanar spacings to be 1.248 and 0.951 Å, respectively.With selected area diffraction, Bragg peak separations, amplitudes, andwidths were obtained as a function of time. The results indicatedifferent timescales from those of image dynamics.

FIG. 6A illustrates structural dynamics and heat dissipation in gold andFIG. 6B illustrates coherent resonance of graphite. Referring to FIG.6A, SADD for fs excitation at 1.7 mJ/cm² peak fluence (519 nm) isillustrated. The Bragg separation for all peaks and the amplitude of the{042} peaks are shown in the main panel; the inset gives the 2.2 μsrecovery (by cooling) of the structure obtained by stroboscopic nsexcitation at 7 mJ/cm². The peak amplitude has been normalized to thetransmitted beam amplitude, and the time dependence of amplitude andseparation is fit as an exponential rise, and a delay with rise,respectively. Referring to FIG. 6B, resonance oscillations are observedfor the Bragg (1 22) peak in the diffraction pattern of graphite; theamplitudes are similar in magnitude to those in FIG. 6A. The sample wastilted at 21° angle to the microscope axis and the diffraction patternwas obtained by using the SAD aperture of 6 μm diameter on the specimen.The graphite thickness is 69 nm as determined by EELS; the oscillationperiod (τ_(p)) is measured to be 56.3 ps. For a thickness of 45 nm, theperiod is found to be τ_(p)=35.4 ps.

FIGS. 6D-G illustrate, for selected areas, time dependence of intensitydifference (dark-field) for graphite. The image change displays theoscillatory behavior with the same τ_(p) as that of diffraction. Thedark-field (DF) images were obtained by selecting the Bragg (1 22) peak.In FIG. 6H, each line corresponds to the difference in imageintensities, Im(t−30 ps; t), for selected areas of 1×100-pixel slicesparallel to contrast fringes in the DF image.

The average amplitude of {042} diffraction peaks drop significantly; therise time is 12.9 ps, whereas the change in separations of all planes isdelayed by 31 ps and rises in 60 ps. The delay in the onset ofseparation change with respect to amplitude change is similar to thetimescale for the amplitude to reach its plateau value of 15% reductionin the case of the {042} amplitude shown. In order to determine therecovery time of the structure, we carried out stroboscopic (and alsosingle-pulse) experiments over the timescale of microseconds. Therecovery transient in the inset of FIG. 6A (at 7 mJ/cm²) gives a timeconstant of 2.2 μs; we made calculations of 2D lateral heat transportwith thermal conductivity (λ=3.17 W/(cm K) at 300 K) and reproduced theobserved timescale. For this fluence, the maximum lattice spacing changeof 0.08% gives the temperature increase ΔT to be 60 K, knowing thethermal expansion coefficient of gold (α=14.2×10⁻⁶ K⁻¹).

The atomic-scale motions, which lead to structural and morphologicalchanges, can now be elucidated. Because the specimen is nanoscale inthickness, the initial temperature induced is essentially uniform acrossthe atomic layers and heat can only dissipate laterally. It is knownthat for metals the lattice temperature is acquired following the largeincrease in electron temperature. The results in FIG. 6A give thetemperature rise to be 13 ps; from the known electron and latticeheat-capacity constants [C₁=70 J/(m³K²) and C₂=2.5×10⁶ J/(m³K),respectively] and the electron-phonon coupling [g=2×10¹⁶ W/(m³K)] weobtained the initial heating time to be ˜10 ps for electron temperatureT₁=2500 K, in good agreement with the observed rise. Reflectivitymeasurements do not provide structural information, but they give thetemperature rise. For bulk material, the timescale for heating (˜1 ps)is shorter than that of the nano-scale specimen (˜10 ps), due toconfinement in the latter, which limits the ballistic motion ofelectrons in the specimen, and this is evident in the UEM studies.Because the plane separation is 0.4078 nm, the change of the averagepeak separation (0.043%), at the fluence of 1.7 mJ/cm², gives a latticeconstant change of 0.17 pm.

Up to 30 ps the lattice is hot but, because of macroscopic latticeconstraint, the atomic stress cannot lead to changes in lateralseparations, which are the only separations visible for the [100]zone-axis probing. However, the morphology warping change is correlatedwith atomic (lateral) displacements in the structure as it relieves thestructural constraint. Indeed the time scale of the initial image changeis similar to that of plane separations in diffraction (60-90 ps). Thisinitial warping, which changes image contrast, is followed by longertime (ns) minimization of surface energy and bulging, as shown in FIG.5D. Given the picometer-scale structural change (0.17 pm), the stressover the 8-nanometer specimen gives the total expansion to be 3.4 pmover the whole thickness. Considering the influence of lateralexpansion, the maximum bulge reaches 1 to 10 nm depending on the lateralscale. Finally, the calculated Debye-Waller factor for structuralchanges gives a temperature of 420 K (ΔT=125 K), in excellent agreementwith lattice temperature derived under similar conditions, noting thatfor the nanoscale material the temperature is higher than in the bulk.

Graphite was another study in the application of the UEM methodology. Incontrast to the dynamics of gold, in graphite, because of its unique 2Dstructure and physical properties, we observed coherent resonancemodulations in the image and also in diffraction. The damped resonanceof very high frequency, as shown below, has its origin in the nanoscaledimension of the specimen and its elasticity. The initial fs pulseinduces an impulsive stress in the film and the ultrafast electrontracks the change of the transient structure, both in SAID and SADD. InFIG. 6B, the results obtained by measuring changes of the diffractionspot (1 22) are displayed and in FIGS. 6D-G those obtained by dark-field(DF) imaging with the same diffraction spot being selected by theobjective aperture and the specimen tilted, as discussed below.

For both the image and diffraction, a strong oscillatory behavior isevident, with a well defined period and decaying envelope. When thetransients were fitted to a damped resonance function [(cos2πt/τ_(p))exp(−t/τ_(decay))], we obtained τ_(p)=56.3±1 ps for theperiod. The decay of the envelope for this particular resonance issignificantly longer, τ_(decay)=280 ps. This coherent transient decay,when Fourier transformed, indicates that the length distribution of thefilm is only ±2 nm as discussed in relation to the equation below. Thethickness of the film was determined (L=69 nm) using electron energyloss spectra (EELS).

In order to test the validity of this resonance behavior we repeated theexperiments for another thickness, L=45 nm. The period indeed scaledwith L, giving τ_(p)=35.4 ps. These, hitherto unobserved, very highfrequency resonances (30 gigahertz range) are unique to the nanoscalelength of graphite. They also reflect the (harmonic) motions due tostrain along the c-axis direction, because they were not observed whenwe repeated the experiment for the electron to be along the [001] zoneaxis. The fact that the period in the image is the same as that of thediffraction indicates the direct correlation between local atomicstructure and macroscopic elastic behavior.

Following a fs pulse of stress on a freely vibrating nanofilm, theobserved oscillations, because of their well-defined periods, arerelated to the velocity (C) of acoustic waves between specimenboundaries, which in turn can be related to Young's modulus (Y) of theelastic stress-strain profile:

$\begin{matrix}{\frac{1}{\tau_{p}} = \frac{n\; C}{2\; L}} \\{{= {\frac{n}{2\; L}\left( \frac{Y}{\rho} \right)^{1\text{/}2}}},}\end{matrix}$where n is a positive integer, with n=1 being the fundamental resonancefrequency (higher n are for overtones). Knowing the measured τ_(p) andL, we obtained C=2.5×10⁵ cm/s. For graphite with the density ρ=2.26g/cm³, Y=14.6 gigapascal for the c-axis strain in the natural specimenexamined. Pyrolytic graphite has Y values that range from about 10 to1000 gigapascal depending on the orientation, reaching the lowest valuein bulk graphite and the highest one for graphene. The real-timemeasurements reported here can now be extended to different lengthscales, specimens of different density of dislocations, andorientations, exploring their influence at the nanoscale on C, Y, andother properties. We note that selected-area imaging was critical asdifferent regions have temporally different amplitudes and phases acrossthe image.

Uniting the power of spatial resolution of EM with the ultrafastelectron timing in UEM provides an enormous advantage when seeking tounravel the elementary dynamics of structural and morphological changes.With total dissipation of specimen heat between pulses, selected-areadynamics make it possible to study the changes in seconds of recordingand for selected pixels of the image. In the applications given here,for both gold and graphite, the difference in timescales for thenonequilibrium temperature (reaching 10¹³ K/s), the structural (pmscale) and morphological (nm scale) changes, and the ultrafast coherent(resonance) behavior (tens of gigahertz frequency) of materialsstructure illustrate the potential for other applications, especiallywhen incorporating the different and valuable variants of electronmicroscopy as we have in our UEM.

Embodiments of the present invention extend ultrafast 4D diffraction andmicroscopy to the attosecond regime. As described herein, embodimentsuse attosecond electron diffraction to observe attosecond electronmotion. Pulses are freely generated, compressed, and tilted. Theapproach can be implemented to extend previous techniques including, forexample phase transformations, chemical reactions, nano-mechanicalprocesses, and surface dynamics, and possibly to other studies ofmelting processes, coherent phonons, gold particles, and molecularalignment.

As described herein, the generation of attosecond resolution pulses andin situ probing through imaging with free electrons. Attoseconddiffraction uses near mono-energetic attosecond electron pulses forkeV-range of energies in free space and thus space charge effects areconsidered. Additionally, spatiotemporal synchronization of the electronpulses to the pump pulses is made along the entire sample area and withattosecond precision. Diffraction orders are shown to be sensitive tothe effect of electron displacement and conclusive of thefour-dimensional dynamics.

A component of reaching attosecond resolution with electron diffractionis the generation of attosecond electron pulses in “free space,” so thatdiffraction from freely chosen samples of interest can take placeindependent of the mechanisms of pulse generation. Electrons withenergies of 30-300 keV are ideal for imaging and diffraction, because oftheir high scattering cross sections, convenient diffraction angles, andthe appropriate de Broglie wavelength (0.02 to 0.07 Å) to resolveatomic-scale changes. Moreover, they have a high degree ofmonochromaticity. For example, electrons accelerated to E₀=30-300 keVwith pulse duration of 20 attoseconds (bandwidth of ΔE≈30 eV) haveΔE/E₀≈10⁻³-10⁻⁴, making diffraction and imaging possible without aspread in angle and resolution. Optical attosecond pulses have typicallyΔE/E₀≈0.5 and because of this reach of ΔE to E₀, their duration isFourier-limited to ˜100 attoseconds. Free electron pulses of keV centralenergy can, in principle, have much shorter duration, down tosub-attoseconds, while still consisting of many wave cycles.

Pulses with a large number of electrons suffer from the effect of spacecharge, which determines both the spatial and the temporal resolutions.This can be avoided by using packets of single, or only a few, electronsin a high repetition rate, as demonstrated in 4D microscopy imaging.FIG. 7A depicts the relation of single electron packets to the effectiveenvelope due to statistics. Each single electron (blue) is a coherentpacket consisting of many cycles of the de Broglie wave and hasdifferent timing due to the statistics of generation. On average,multiple single electron packets form an effective electron pulse(dotted envelope). It will be appreciated that there is high dispersionfor electrons of nonrelativistic energy. The small but unavoidablebandwidth of an attosecond electron pulse causes the pulse to disperseduring propagation in free space, even when no space charge forces arepresent. For example, a 20-attosecond pulse with ΔE/E₀≈10⁻³ wouldstretch to picoseconds after just a few centimeters of propagation.

Embodiments of the present invention provide methods and systems for thesuppression of dispersion and the generation of free attosecond electronpulses based on the initial preparation of negatively-chirped electronpackets. As described herein, femtosecond electron pulses are generatedby photoemission and accelerated to keV energies in a static electricfield. Preceding the experimental interaction region, optical fields areused to generate electron packets with a velocity distribution, suchthat the higher-energy parts are located behind the lower-energy ones.With a proper adjustment of this chirp, the pulse then self-compressesto extremely short durations while propagating towards the point ofdiffraction. To achieve attosecond pulses, the chirp must be imprintedto the electron pulse on a nanometer length scale. Optical waves providesuch fields. However, non-relativistic electrons move significantlyslower than the speed of light (e.g. ˜0.3 c for 30 keV). The directinteraction with an optical field will, therefore, cancel out over timeand can not be used to accelerate and decelerate electrons forcompression. In order to overcome this limitation, we make use of theponderomotive force, which is proportional to the gradient of theoptical intensity to accelerate electrons out of regions with highintensity. By optical wave synthesis, intensity profiles can be madethat propagate with less than the speed of light and, therefore, allowfor co-propagation with the electrons.

FIG. 7B illustrates a schematic of attosecond pulse generation accordingto an embodiment of the present invention. A synthesized optical fieldof two counter-propagating waves of different wavelengths results in aneffective intensity grating, similar to a standing wave, which moveswith a speed slower than the speed of light. Electrons can, therefore,co-propagate with a matched speed and are accelerated or decelerated bythe ponderomotive force according to their position within the wave.After the optical fields have faded away, this velocity distributionresults in self-compression; the attosecond pulses are formed in freespace. Depending on the optical pulse intensity, the electron pulseduration can be made as short as 15 attoseconds, and, in principle,shorter durations are achievable. If the longitudinal spatial width ofthe initial electron pulse is longer than the wavelength of theintensity grating, multiple attosecond pulses emerge that are locatedwith well-defined spacing at the optical minima. This concept ofcompression can be rigorously described analytically as a “temporal lenseffect.” The temporal version of the Kapitza-Dirac effect has aninteresting analogy.

Some of our initial work was based on an effective ponderomotive forcein a collinear geometry. In order to extend the approach to more complexarrangements, here we generalize the approach and consider the fullspatiotemporal (electric and magnetic) fields of two colliding laserwaves with an arbitrary angle and polarization. The transversal andlongitudinal fields of a Gaussian focus were applied. We simulatedelectron trajectories by applying the Lorentz force with a fourth-orderRunge-Kutta algorithm using steps of 100 attoseconds. Space chargeeffects were taken into account by calculating the Coulomb interactionsbetween all single electrons for each time step (N-body simulations).

FIG. 7B illustrates temporal optical gratings for the generation of freeattosecond electron pulses for use in diffraction. (a) A femtosecondelectron packet (blue) is made to co-propagate with a moving opticalintensity grating (red). (b) The ponderomotive force pushes electrontowards the minima and thus creates a temporal lens. (c) The inducedelectron chirp leads to compression to attosecond duration at latertime. (d) The electron pulse duration from 10⁵ trajectories reaches intothe domain of few attoseconds.

FIG. 7C depicts the compression of single electron packets in thecombined field of two counter-propagating laser pulses with durations of300 fs at wavelengths of 1040 nm and 520 nm. The pulse is shown justbefore, at, and after the time of best compression; the center along Zis shifted for clarity. The plotted pulse shape is a statistical averageover 10⁵ packets of single electrons. The beam diameter of the initialelectron packet was 10 μm and the beam diameters of the laser pulseswere 60 μm; the resulting compression dynamics is depicted before, justat, and some time after the time of best compression to a duration of 15attoseconds (see FIG. 7C(b)). These results show that an optical wavewith a beam diameter of only several times larger than that of theelectron packet is sufficient to result in almost homogeneouscompression along the entire electron beam. The characteristiclongitudinal spread after the point of best compression, as depicted inFIG. 7C is the result of an “M”-shaped energy spectrum of the electronsafter interactions with the sinusoidal intensity grating.

Coulomb forces prevent concentration of a large number of electrons in alimited volume, and a compromise between electron flux and laserrepetition rate must be found to achieve sufficiently intensediffraction. The laser pulses for compression have energies on the orderof 5 μJ and can, therefore, be generated at MHz repetition rates withthe resulting flux of 10⁶ electrons/s, which is sufficient forconclusive diffraction. Nevertheless, having more than one electron perattosecond pulse is beneficial for improving the total flux.

In order to investigate the influence of space charge on the performancein our attosecond compression scheme, we considered electron packets ofincreasing electron density and evaluated the resulting pulse durationsand effective electron density per attosecond pulse. Two findings arerelevant with the results shown in FIG. 8. First, the duration ofindividual attosecond electron pulses increases relativelyinsignificantly with the number of electrons contained within. Even for40 electrons in a single pulse, the duration increases only from 15 to25 attoseconds (see FIG. 8( a)). The reasons for this are the highlyoblate shape of the electron pulses, and the approximate linearity ofspace charge forces in the longitudinal direction, which are compensatedfor by somewhat longer interaction in the ponderomotive forces of theoptical waves.

Secondly, for a train of pulses, there is an effect on synchronization.When the initial femtosecond electron packet covers several opticalcycles of the compression wave, a train of attosecond pulses results asshown in FIG. 7B. Perfect synchronization to the optical wave isprovided, because all attosecond pulses are located at the same opticalphase of the fundamental laser wave. This phase matching relation, whichpermits attosecond resolution, despite the presence of multiple pulses,is altered under space charge conditions. The attosecond pulses repeleach other and a temporal spreading of the comb-like train results. Fora train of near 10 attosecond pulses, FIG. 8( b) displays the differencein timing for an adjacent attosecond pulse in relation to the centralone, which is always locked to the optical phase because the spacecharge forces cancel out. The total timing mismatch is the product ofthe plotted value with the number of attosecond pulses in the entireelectron packet (near 10 for this example). In order to keep the totalmismatch to the optical wave below 20 attoseconds, 10 electrons perattosecond electron pulse represent an optimum value. The total pulsetrain then consists of 200 electrons for that group of pulses; of coursethe total flux of electrons is determined by the repetition rate. Notethat mismatch to the compression wave is absent with isolated attosecondelectron pulses, which are generated when the initial uncompressedelectron packet is shorter than a few femtoseconds, or with opticalfields of longer wavelength.

Numerous imaging experiments have been successful with single electronpackets. In state-of-the-art electron crystallography experiments,typically 500 electrons per pulse were used at a repetition rate of kHzto produce the needed diffraction. This is equivalent to having 5electrons per attosecond pulse at 100 kHz, which is a convenientrepetition rate for optical wave synthesis, and provides enough time forletting the material under investigation to cool back to the initialstate. Laser systems with MHz repetition rates will provide even higherfluxes.

Applications of attosecond electron pulses for diffraction andmicroscopy use synchronization of events in the pump-probe arrangementwith an accuracy that is equal or better than the individual pulsedurations. In contrast to recompression concepts that are based ontime-dependent microwave fields, the application of laser waves forattosecond electron pulse generation provides exact temporalsynchronization when the pump pulses are derived by phase-locking fromthe same laser system. Many common optical techniques, such as nonlinearfrequency conversion, continuum generation in solids, or high-harmonicgeneration, all provide a phase lock in the sense that the outcome hasthe same relative phase and timing in relation to the incoming opticalwave for each single pulse of the laser.

A second requirement for reaching into the temporal resolution ofattoseconds is the realization of spatial delay matching along extendedareas of the diffraction. The use of large samples, for example with upto millimeters in size in some electron diffraction experiments,provides enhanced diffraction efficiency and offers the possibility touse electron beams with large diameter, in order to maximize thecoherence and flux. In this case, the time resolution is limited bydifferences in the arrival times of pump and probe pulses at differentpoints within the involved beam diameters (group velocity mismatch).

Electron pulses at keV energies travel with significantly less than thespeed of light (e.g. v_(el)=0.3 c for 30 keV electrons) and are,therefore, “overtaken” by the laser wave. Embodiments of the presentinvention provide two arrangements for matching the group velocity ofelectrons with the phase velocity of optical pulses. Both arrangementsare suitable for applications in noncollinear, ultrafast electronmicroscopy and diffraction. FIG. 9( a) presents a concept for thetransmission geometry of diffraction and microscopy in which two anglesare introduced, one between the laser beam and the electron beam (β),and another one (α) for the tilt angle of the sample (black) to thephase fronts of the laser wave. Total synchrony is achieved if therelative delay between the optical wave and the attosecond electronpulses is made identical for all points along the entire sample surface.Each small volume of the sample is then subject to an individualpump-probe-type experiment with the same time delay.

The above condition is found when we match the coincidence along theentire width of the specimen. The effective surface velocity v_(surface)of the laser and of the electron pulses must be identical. From FIG. 9A,this requirement is expressed by the following equation:

$\begin{matrix}{\frac{\sin(\alpha)}{\sin\left( {\alpha - \beta} \right)} = {\frac{c}{v_{el}}.}} & (1)\end{matrix}$

It follows that an angle of β=10°, for example, results in an optimumangle for the sample tilt of α=14.8°, which are both easily achievableangles in a real experiment. The effective tilt of the sample withrespect to the electron direction is then α−β=4.8°. Naturally, if thisvalue is not coincident with a zone axis direction, a complete rockingcurve should be obtained in order to optimize α and β with tiltrequirements. Although different portions of the laser wavefront impingeon the surface of the sample at different times, this behavior ismatched by the electron pulse, resulting in all portions of the surfaceof the sample being phase matched.

As illustrated in FIG. 9( a), the laser beam, also referred to as alaser wave, is used to activate the sample, for example, to heat thesample, cause motion of the sample, or to effect the chemical bondspresent in the sample. The timing of the laser wave and the electronpulses are synchronized using the delay stage discussed in relation toFIG. 1. The train of electron pulses can be generated using theconfiguration illustrated in FIG. 10( a).

Another option for synchronization along extended surfaces is the use oftilted electron pulses, for that the electron density makes an anglewith respect to the propagation direction. Tilted optical pulses havebeen used for reaching femtosecond resolution in reflection geometry,but here tilted electron pulses are introduced for effectivespatiotemporal synchronization to the phase velocity of the excitationpulses along the entire sample surface. FIG. 9B depicts the concept. Ifan angle γ is chosen between the laser (red) and the attosecond electronpulses (blue), the electron pulses need to be tilted likewise. Thesample is located parallel to the optical phase fronts and its entiresurface is illuminated by the attosecond electron pulse at once and atthe same time of incidence relative to the optical pulse wave. Becausethe incidence is delay-free for all points along the surface, velocitymatching is provided for the whole probed area.

The generation of tilted attosecond electron pulses is outlined in FIG.10( a). The introduction of an angle between the intensity grating (red)and the electron beam (blue) leads to formation of electron pulses witha tilt. As described above, a femtosecond electron packet (blue) isfirst generated by conventional photoelectron generation and acceleratedin a static electric field. By intersecting the counter-propagatingintensity grating at an angle, tilted electron pulses result withattosecond duration. The ponderomotive force accelerates the electronstowards the planes of destructive interference in the intensity wave andthey form attosecond pulses that are compressed along the optical beamaxis; but the pulses propagate in the original direction. Only a slightadjustment of the electrons' central energy is required to achieve phasematching to the moving optical grating.

Based on this concept, we simulated the tilting effect by using 31-keVelectron pulses with an initial duration of ˜15 femtoseconds and aspatial beam diameter of ˜10 μm. FIG. 10( b) illustrates the simulationresults for an initial packet of 15-femtosecond duration (left) and anintersection angle of 5°. The tilted attosecond pulses have duration of˜20 attoseconds when measured perpendicular to the tilt (note thedifferent scale of Z and X). The optical intensity wave is synthesizedby two counter-propagating laser pulses of 100-fs duration andwavelengths of 1040 and 520 nm. The angle between the electron beam andthe laser wave is 5°. The results of compression are shown in FIG. 10(b): The attosecond electron pulses are formed at the minima of theoptical intensity wave and, therefore, are tilted by 5° with respect tothe electron propagation direction. For other incidence angles of thelaser, the electron pulses are tilted accordingly. Perpendicular to theattosecond pulses, the measured duration is ˜20 attoseconds, given asthe full width at half maximum.

Based on the methodology for generation and synchronization ofattosecond electron pulses described above, the diffraction andmanifestation of electron dynamics in the patterns are described. By wayof two different examples, embodiments of the present invention areutilized to observe electronic motions in molecules and materials withattosecond electron packets.

We consider first the physics of electron scattering and the change inscattering factors which characterize individual atoms and the electrondensity involved. Diffraction from molecular crystals or othercrystalline structures provides two distinct advantages over thatobtained for gas phase ensembles. First, the sample density is manyorders of magnitudes higher (10²¹ molecules/cm³ as compared to 10¹⁰ to10¹⁶/cm³ in gas jets); diffraction is, therefore, more intense. Second,the crystalline order results in Bragg scatterings and they areconcentrated into well-defined “spots” for ordered crystals; thepatterns become rods for surfaces and narrow rings for amorphoussubstances. The diffraction results in intensities which areproportional to the square of the diffraction amplitude. As discussedbelow, coherence in diffraction is used in observing the changes ofinterest.

The diffraction from molecular crystals, or other crystalline materialsof interest, is defined by the summation over the contributions of allscatterers in a unit cell. The intensity I of a Bragg spot with theMiller indices (hkl) is determined by the positions (xyz) of thescatterers j in the unit cell:

$\begin{matrix}{{{I({hkl})} \propto {{\sum\limits_{j}^{\;}\;{f_{j}{\exp\left\lbrack {{- 2}\;\pi\;{{{\mathbb{i}}({hkl})} \cdot ({xyz})_{j}}} \right\rbrack}}}}^{2}},} & (2)\end{matrix}$where f_(j) are the atomic scattering factors. Electron diffraction isthe result of Coulomb interaction between the incoming electrons and thepotential formed by nuclei and electrons. The factors f_(j) account forthe effective scattering amplitude of atoms and are derived from quantumcalculations that take into account the specific electron densitydistribution around the nuclei, including core electrons. The scatteringwe are considering here is the elastic one.

In order to estimate the influence of electron dynamics on contributionsto time-resolved diffraction patterns, we consider typical densities ofelectrons in chemical bonds, and the possible change. Static electrondensity maps show that typical covalent bonds consist of about oneelectron/Å³ and that this density is delocalized over volumes withdiameters in the order of 1 Å. For estimating an effective scatteringfactor of such electron density, we consider a Gaussian sphere with adiameter of 1 Å, consisting of one electron. The electric potential isderived by Gauss' law and results in a radial dependence that isrepresented in FIG. 11, dotted line. The total scattering amplitude offree charges diverges at small angles, because of the long-rangebehavior of the potential. Since in real crystals the potential islocalized in unit cells, we use a Gaussian distribution of the samemagnitude in order to restrict the range to about ±1.5 Å.

For potential of spherical symmetry, an effective scattering factor canbe calculated from the radial potential Φ(r) according to

$\begin{matrix}{{{f_{el}(s)} = {\frac{8\;\pi^{2}m_{e}e}{h^{2}}{\int_{0}^{\infty}{r^{2}{\Phi(r)}\ \frac{\sin\left( {4\;\pi\;{sr}} \right)}{4\;\pi\;{sr}}{\mathbb{d}r}}}}},} & (3)\end{matrix}$where s=sin(θ/2)/λ_(el) is the scattering parameter for a diffractionangle θ and λ_(el) is the de Broglie wavelength of the incidentelectrons. The result for our delocalized electron density is shown inFIG. 11( b); for comparison we plot also the tabulated scattering factorof neutral hydrogen. Both have comparable magnitude, which is expectedbecause of their similar sizes.

Here, we consider the iodine molecule as a model case and invoke thetransition from a bonding to an anti-bonding orbital. The crystalstructure of iodine consists of nearly perpendicular iodine pairs with abond length of ˜2.7 Å. Two electrons contribute to the intramolecular σbond; the intermolecular bond is weaker.

FIG. 12 depicts the system under study and the two cases considered. Theeffect of antibonding excitation is made by comparing the Braggintensities for the iodine structure, including the binding electrons,to a hypothetical iodine crystal consisting only of isolated atoms (seeFIG. 12( a)). In Table 1, we give the results of the calculationsfollowing equation 2 with the values of f tabulated for iodine atoms andfrom equation (3) for the electronic distribution changes. Despite thelarge difference in f of the iodine nuclei and the electron (about50:1), the changes of Bragg spot intensity are significant, being on anorder of 10-30%.

TABLE 1 Effects of Electron Motion on Selected Molecular Bragg SpotsMiller Indices (hkl) (a) ΔI_(Transition) (b) ΔI_(Movement)(0.08 Å) 100,010, 001 (forbidden) (forbidden) 200, 400, 600 0 0 002  −35% 0 020(weak) +100% −17% 400 0 0 040  −18% +13% 004 0 0 111  +15%  −2% 331 −20% +15%

In column (a), the m Magnitude of Bragg spot intensity change ΔI ofcrystalline iodine as a result of bonding to antibonding transition isgiven. In column (b), the magnitude of Bragg spot intensity change as aresult of field interaction with charge density, also in iodine.

This large change is for two reasons. First, the bonding electrons arelocated in-between iodine atoms and contribute, therefore, strongly tothe enhancement or suppression of all Bragg spots that project from theinter-atomic distances of the molecular units. Second, the large effectis result of the intrinsic “heterodyne detection” scheme of diffraction;the total intensity of a Bragg spot scales with the square of thecoherent sum of individual contributions (see equation (2)). Althoughthe total contribution to the intensity of a Bragg spot from bondingelectrons is lower by a factor of several hundreds than the intensitycontributions from the iodine atoms, the modulation is on the order ofseveral percents as a result of the coherence of diffraction on ananometer scale. Symmetry in the crystal is evident in the absence ofchange in certain Bragg orders. From measurements of the dynamics ofmultiple spots, it follows that electron density movies could be made.This is best achieved in an electron microscope in diffraction geometry;however conventional diffraction is also suitable to simultaneouslymonitor many Bragg spots and is advantageous for the study of orderedbulk materials. The example given is not far from an experimentalobservation made on a metal-to-insulator transition for which a σ*-typeexcitation was induced with a femtosecond pulse.

As a second model case we consider the reaction of bonded electrondensity to external electric fields, such as the ones from laser fields.Depending on the restoring force and the resonance, an electron densitywill oscillate with the driving field in phase or with a phase delay.This charge oscillation re-radiates and is responsible for therefractive index of a dielectric material. In order to estimate themagnitude of charge displacement, we must take into account thepolarizability, α, and the electric field strength, E_(laser). In thelimit of only one moving charge, the displacement D is approximatelygiven by

$D \approx {\frac{\alpha}{e}{E_{laser}.}}$

The polarizability of molecular iodine along the bond is α≈130 ∈₀ Å³(˜70 a.u.) in the static limit and a similar magnitude is expected forthe crystal for optical frequencies away from the strong absorptionbands; the anisotropy of polarizability indicates the role of thebonding electrons. With femtosecond laser pulses, a field ofE_(laser)=10⁹ V/m is possible for intensities below the damagethreshold. With these parameters, one expects a charge displacement ofD≈0.08 Å, or about 3% of the bond length.

FIG. 12( b) is a schematic for the change in charge distribution by anelectric field. We assume an active role of only the bonded electrons,and take the polarization of the laser field to be along the b axis ofsolid iodine. This axis is chosen because it has the least symmetry; ais perpendicular to the bonds. Table 1 gives the intensity changes ofselected Bragg spots; the change is in the range of ±20% for some of theindices. The total energy delivered to the molecular system by the laserfield is only on the order of 0.01 eV. Nevertheless the changes ofcharge displacements on sub-angstrom scales are evident. This marks acentral advantage of electron diffraction over spectroscopic approaches,which require large energy changes in order to have projections ondynamics. In contrast, diffraction allows for the direct visualizationof a variety of ultrafast electron dynamics with combined spatial andtemporal resolutions, and independent of the resolution of internalenergy levels.

The “temporal lens” concept can be used for the focus and magnificationof ultrashort electron packets in the time domain. The temporal lensesare created by appropriately synthesizing optical pulses that interactwith electrons through the ponderomotive force. With such anarrangement, a temporal lens equation with a form identical to that ofconventional light optics is derived. The analog of ray diagrams, butfor electrons, are constructed to help the visualization of the processof compressing electron packets. It is shown that such temporal lensesnot only compensate for electron pulse broadening due to velocitydispersion but also allow compression of the packets to durations muchshorter than their initial widths. With these capabilities ultrafastelectron diffraction and microscopy can be extended to new domains, but,as importantly, electron pulses are delivered directly on the targetspecimen.

With electrons, progress has recently been made in imaging structuraldynamics with ultrashort time resolution in both microscopy anddiffraction. Earlier, nuclear motions in chemical reactions were shownto be resolvable on the femtosecond (fs) time scale using pulses oflaser light, and the recent achievement of attosecond (as) light pulseshas opened up this temporal regime for possible mapping of electrondynamics. Electron pulses of femtosecond and attosecond duration, ifachievable, are powerful tools in imaging. The “electron recombination”techniques used to generate such attosecond electron pulses require theprobing electron to be created from the parent ions (to date noattosecond electron pulses have been delivered on an arbitrary target)and for general applications it is essential that the electron pulse bedelivered directly to the specimen.

In ultrafast electron microscopy (UEM), the electron packet duration isdetermined by the initiating laser pulse, the dispersion of the electronpacket due to an initial energy spread and electron-electroninteractions. Since packets with a single electron can be used to image,and the initiating laser pulse can in principle be made very short(sub-10 fs), the limiting factor for the electron pulse duration is theinitial energy spread. In photoelectron sources this spread is primarilydue to the excess energy above the work function of the cathode, and isinherent to both traditional photocathode sources and optically-inducedfield emission sources. Energy-time uncertainty will also cause ameasurable broadening of the electron energy spread, when the initiatinglaser pulse is decreased below ˜10 fs. For ultrafast imaging techniquesto be advanced into the attosecond temporal regime, methods fordispersion compensation and new techniques to further compress electronpulses to the attosecond regime need to be developed.

As described herein techniques for compressing free electron packets,from durations of hundreds of femtoseconds to tens of attoseconds, usingspatially-dependent ponderomotive potentials are provided by embodimentsof the present invention. Thus, a train of attosecond pulses can becreated and used in ultrafast electron imaging. Because they aregenerated independent of the target they can be delivered to a specimenfor studies of transient structures and electronic excitations on theattosecond time scale. The deflection of electrons (as in theKapitza-Dirac effect) by the ponderomotive potential of intense lasersand the diffraction of electrons in standing waves of laser light havebeen observed, and so is the possibility (described through computermodeling) of spatial/temporal focusing with combined time-dependentelectric and static magnetic fields.

The “temporal lens” description analytically expresses how ponderomotivecompression can be used to both compensate for the dispersion andmagnify, in this case compress, the temporal duration of electronpackets. We obtain simple lens equations which have analogies in opticsand the results of “electron ray optics” of temporal lenses allows foranalytical expressions and for the design of different schemes usinggeometrical optics. Here, we consider two types of temporal lenses, thinand thick.

For the realization of the temporal thin lens, a laser beam with aLaguerre-Gaussian transverse mode, radial index ρ=0 and azimuthal indexl=0 (or, in common nomenclature, a “donut” mode, is utilized. In thecenter of the donut mode, electrons will experience a spatially-varyingponderomotive potential (intensity) that is approximately parabolic.This potential corresponds to a linear spatial force which, for chirpedelectron pulses, can lead to compression from hundreds of fs to sub-10fs. The second type, that of a thick lens, is based on the use of twocounter-propagating laser beams in order to produce aspatially-dependent standing wave that co-propagates with the electrons.A train of ponderomotive potential wells are produced at the nodes ofthe standing wave, leading to compression but now with much “tighterfocus” (thick lens). Because the electron co-propagates with the laserfields, velocity is matched. Analytical expressions are derived showingthat this lens has the potential to reach foci with attosecond duration.Finally, we discuss methods for creating tunable standing waves forattosecond pulse compression, and techniques for measuring the temporaldurations of the compressed pulses. Space-charge dispersed packets ofelectrons that have a linear spatial velocity chirp may also becompressed with the temporal lenses described here.

All electron sources, both cw and pulsed, have an initial energy spread.For pulsed electron sources this is particularly relevant as electronpackets created in a short time disperse as they propagate. The initialenergy spread leads to an initial spread in velocities. These differentvelocities cause the initial packet to spread temporally, with thefaster electrons traveling a further distance and the slower electronstraveling a shorter distance in a given amount of time. The dispersionleads to a correlation between position (along the propagationdirection) and electron velocity as described in relation to FIG. 14.The linear spatial velocity “chirp” can be corrected for with aspatially-dependent linear impulsive force (or a parabolic potential).Thus, if a pulsed, spatially-dependent parabolic potential can be madeto coincide appropriately with the dispersed electron packet, the slowtrailing electrons can be sped up and the faster leading electrons canbe slowed down. The trailing electrons, now traveling faster, can catchthe leading electrons and the electron pulse will thus be compressed.

FIG. 13 illustrates dispersion of an ultrashort electron packet. Att=t_(o) the packet is created from a photocathode and travels with avelocity ν₀. As it propagates along the x-axis it disperses, with thefaster electrons traveling further, and the slower ones trailing for agiven propagation time t. At t=0 a parabolic potential is pulsed on,giving an impulsive “kick” to the dispersed electron packet. After thepotential is turned off, t>τ, the trailing electrons now have a greatervelocity than the leading electrons. After a propagation time t=t_(i),the pulse is fully compressed.

Consider a packet of electrons, propagating at a speed ν₀ along thex-axis, with a spread in positions of Δx_(o)=ν₀Δt_(o), at time t=t_(o).At t=0, a potential of the form U(x)=½Kx² interacts with the electronpacket for a duration τ in the lab frame. The waist, or spatial extentof the potential (temporal lens) is chosen to be w, while the duration τis chosen such that it is short compared to w/ν₀. When this condition ismet the impulse approximation holds, and the change in velocity isΔν=−τ/m(dU(x)/dx)=−τKx/m, for |x|<w, where m is the electron mass. Afterthe potential is turned off, t>τ, the electrons will pass through thesame position, x_(f)−x=(ν₀+Δν)t_(f), at the focal timet_(f)=−x/Δν=m/(Kτ). To include an initial velocity spread around ν₀ (dueto an initial ΔE), consider electrons that all emanate from a sourcelocated at a fixed position on the x-axis. An electron traveling exactlyat ν₀ will take a time t₀ to reach the center of the potential well atx=0. Electrons leaving the source with other velocities ν₀+ν_(k) willreach a location x=ν_(k)t_(o) at t=0. The image is formed at a locationwhere electrons traveling with a velocity ν₀ and a velocity ν₀+ν_(k)intersect, this is, when ν₀t_(i)=x+(ν₀+Δν+ν_(k))t_(i). The image timet_(i) is then t_(i)=−x/(Δν+ν_(k)).

FIG. 14 illustrates ray diagrams for spatial and temporal lenses. Thetop figure in FIG. 14 depicts three primary rays for an optical thinspatial lens. The object is located at y_(o), and the spatial lens has afocal length, f. A real image of the object is created at the imageplane, position y_(i). The bottom figure in FIG. 14 is a ray diagram fora temporal thin lens. The diagram is drawn in a frame moving with theaverage speed ν₀ of the electron packet. The slopes of the differentrays in the temporal diagram correspond to different initial velocitiesthat are present in the electron packet. As shown in the diagram, atemporal image of the original electron packet is created at the imagetime t_(i). The initial packet (object) is created at a time t_(o) withΔt_(o)=Δx_(o)/ν₀, where the spatial extend of the pulse is directlyrelated to the temporal duration of the object. The lens is pulsed on att=0 and the temporal focal length of the lens is t_(f). The lensrepresents the ponderomotive potential and in this case is on for thevery short time τ.

For the object time, t_(o)=x/ν_(k), image time t_(i)=−x/(Δν+ν_(k)) andthe focal time t_(f)=−x/Δν, the temporal lens equation holds,

$\begin{matrix}{{\frac{1}{t_{o}} + \frac{1}{t_{i}}} = {\frac{1}{t_{f}}.}} & (4)\end{matrix}$

Ray tracing for optical lenses is often used to visualize how differentray paths form an image, and is also useful for visualizing how temporallenses work as shown in FIG. 14. As derived in later sections, themagnification M is defined as the ratio of the electron pulse duration(Δt_(i)) at the image position to the electron pulse duration (Δt_(o)),and is directly proportional to the ratio of the object and image times(−t_(i)/t_(o)) and distances (−x_(i)/x_(o)).

In polar coordinates, a Laguerre-Gaussian (LG₀ ¹) mode has a transverseintensity profile given by, I(r,φ)=I₀exp(1)2r²exp(−2(r/w)²)/w² where wis the waist of the focus and I₀ the maximum intensity. This “donut”mode has an intensity maximum located at r=√{square root over (2)}w/2with a value of I₀=2E_(P)(√{square root over (ln 2/π³)}/(w²τ) whereE_(P) is the energy of the laser pulse and τ is thefull-width-at-half-maximum of the pulse duration, assuming a Gaussiantemporal profile given by exp(−4 ln 2(t/τ)²). The ponderomotive energyU_(P)(x) is proportional to intensity,

$\begin{matrix}{{{U_{P}(x)} = {{{\frac{1}{2}\left\lbrack {\frac{e^{2}\lambda^{2}{\exp(1)}I_{0}}{2\;\pi^{2}m\; ɛ_{0}c^{3}w^{2}}\sqrt{\frac{\ln\; 2}{\pi}}} \right\rbrack}x^{2}} \equiv {\frac{1}{2}K\; x^{2}}}},} & (5)\end{matrix}$where m is the electron mass, e is the electron charge and λ the centralwavelength of the laser radiation and replacing r with x. Near thecenter of the donut mode focus (or x<<w) the intensity distribution isapproximately parabolic, and hence the ponderomotive energy near thedonut center is also parabolic. In analogy with a mechanical harmonicoscillator, the quantity in the square brackets of equation (5) can bereferred to as the stiffness K; it has units of J/m²=N/m, and at 800 nmhas the numerical value of, K≈3.1×10⁻³⁶E_(P)/(w⁴τ). For this parabolicapproximation to be applicable, the spatial extent of the dispersedelectron pulse, at t=0, Δx(0)=ν₀Δt_(o)+Δν_(o)t_(o) must be much smallerthan the laser waist, where the object velocity spread isΔν_(o)=ΔE/√{square root over (2mE)}.

The effect of this parabolic potential on an ensemble of electronsemitted from a source will now be analyzed. The velocity distribution ofthe ensemble is centered around ν₀, with an emission time distributioncentered on −t_(o), where all electrons are emitted from the samelocation x_(o)=−ν₀t_(o). Assuming a single donut-shaped laser pulse isapplied at t=0, and centered at x=0, the electron ensemble is theninfluenced by the potential U(x)=½Kx². The k^(th) electron in theensemble has an initial velocity ν₀+ν_(k) and emission time−t_(o)+t_(k). Using a Galilean transformation to a frame moving withvelocity ν₀, the propagation coordinate x (lab frame) is replaced withthe moving frame coordinate

=x−ν₀t. At t=0 the potential exists for the ultrashort laser pulseduration τ, giving the electron an impulse (or “kick”) dependent on itsinstantaneous position in the parabolic potential. In both frames, theposition of the electron at t=0 is x_(k)(0)=

(0)≡−ν₀t_(k)+ν_(k)t_(o)−ν_(k)t_(k), where x_(k)(t) and

(t) are in the lab and moving frames, respectively. Using the impulseapproximation the electron trajectory immediately after the potential isturned off becomes,

(t)=ν_(k) t+

(0)(1−t/t _(f)),  (6)where t_(f)=m/(Kτ) is the focal time. The electron trajectories, beforeand after t=0, can be plotted in both frames to give the equivalent of aray diagram as illustrated in FIG. 15. Electrons emitted at the sametime, i.e. t_(k)=0, but with different velocities, will meet at theimage position,

=0 in the moving frame at the image time t_(i). The image time is foundby setting

(t_(i))=0, from equation (6), with t_(k)=0,

(t_(i))=ν_(k)t_(i)+ν_(k)t_(o)(1−t_(i)/t_(f))=0 which is equivalent tothe lens equation, equation (4): t_(o) ⁻¹+t_(i) ⁻¹=t_(f) ⁻¹.

An expression for the magnification can be obtained when electrons thatare emitted at different times t_(k) and different velocities ν_(k) areconsidered. If the magnification is defined as M=−t_(i)/t_(o) then thetemporal duration at the image time becomes,Δt _(i) =MΔt _(o),  (7)where Δt_(o) and Δt_(i) are the duration of the electron packet at theobject and image time, respectively. Durations achievable with a thintemporal lens follow from equation (7).

An experimentally realistic temporal lens would use a 50 fs, 800 nmlaser pulse with 350 μJ energy, focused to a waist of w=25 μm. Thesevalues result in a stiffness of K=5.5×10⁻⁸ N/m and a focal time oft_(f)=0.3 ns; t_(f)=m/(Kτ). If the lens is applied 10 cm from thesource, electrons emitted at ν₀=c/10 (3 keV) would have an object timeof t_(o)=x_(o)/ν₀=0.1/(c/10)=3.0 ns. Using the temporal lens equation,equation (4), t_(i) is obtained to be 0.33 ns. Hence, a magnification ofM=−t_(i)/t_(o)=0.1. Consequently, a thin temporal lens can compress anelectron packet with an initial temporal duration of Δt_(o)≈100 fs,after it has dispersed, to an image duration of Δt_(i)≈10 fs. While theexample presented here is for 3 keV electrons, the thin lensapproximation holds for higher energy electrons as long as is chosen tobe short compared to w/ν₀. Experimentally, the thin temporal lens can beutilized in ultrafast diffraction experiments which operate at kHzrepetition rates with lasers that typically possess power that exceedsthe value needed for the ponderomotive compression.

Referring to FIG. 15, thin lens temporal ray diagrams for the lab andco-propagating frames are illustrated. The upper left panel is a raydiagram drawn in the lab frame showing how different initial velocitiescan be imaged to a single position/time. The gray lines are raysrepresenting electrons with different velocities. The lower left panelis a ray diagram drawn in a frame moving with the average velocity ν₀ ofthe electron packet. The rays represent velocities of ν₀/67, ν₀/100 and0. In the co-propagating frame, the relationship between Δt_(o) andΔt_(i) can be visualized as Δt_(i)=−Δt_(o)t_(i)/t_(o). One majordifference between the lab frame and the moving frame is that in thelatter the position of the object and image are moving. The linesrepresenting the object and the image positions are drawn with slopes of−ν₀. The upper right panel depicts the experimental geometry for theimplementation of a thin temporal lens. Note that the laser pulse andelectron packet propagate perpendicular to each other, and that theinterception point between the electrons and photons is at x=0 and t=0.The lower right panel shows how the parabolic (idealized) potentialcompares to the experimentally realizable donut potential. The coloreddots indicate the position of electrons following the rays indicated inthe left bottom diagram.

Above, it was analytically shown that free electron packets can becompressed from hundreds to tens of femtoseconds using a temporal thinlens, which would correspond to a magnification of ˜0.1. Co-propagatingstanding wave can be created by using two different optical frequencies,constructed by having a higher frequency (ω₁) optical pulse traveling inthe same direction as the electron packet and a lower frequency (ω₂)traveling in the opposite direction. When the optical frequencies ω₁,ω₂, and the electron velocity ν₀ are chosen according toν₀=c(ω₁−ω₂)/(ω₁+ω₂), a standing wave is produced in the rest frame ofthe electron as illustrated in FIG. 16. If the electron has a velocityν₀=c/3, and ω₁=2ω₂ then the co-propagating standing wave has aponderomotive potential of the form,

$\begin{matrix}{{{U_{P}(x)} = {\frac{1}{2}\left( \frac{e^{2}E_{0}^{2}}{8\;\pi^{2}{mc}^{2}} \right){\cos^{2}{()}}}},} & (8)\end{matrix}$where E₀ is the peak electric field,

the Doppler shifted wavelength. The envelopes of the laser pulses areignored in this derivation, but they can be engineered so that thestanding wave contrast is optimized. The standing waves can be providedoutside the microscope housing or inside the microscope housing. Thepresence of the standing wave copropagating with the electron pulse orpacket inside the microscope housing can produce a series of attosecondelectron pulses as illustrated in FIG. 7B and FIG. 16. Depending on thegeometry with which the laser beams interact, the standing wave and theelectron pulse can overlap adjacent to the sample, providing attosecondelectron pulse generation at distances close to the sample. Theattosecond electron pulses can be single electron pulses.

To find an analytic solution in the thick lens geometry, each individualpotential well in the standing wave is approximated by a parabolicpotential that matches the curvature of the sinusoidal potential,U_(P)(x)=½[e²E₀ ²/(2mc²)]x²≡½Kx². Using the exact solution to theharmonic oscillator the focal time is,t _(f)=cot(ω_(P)τ)/ω_(P)+τ,  (9)where ω_(P)=√{square root over (Km)} and τ is the duration that the lensis on. For τ→0, t_(f)→m/(Kτ), which is identical to the thin lensdefinition. The image time, t_(i), has a form,t _(i)=(1/ω_(P) ² +t _(o) t _(f) −t _(f)τ+τ²)/(t _(o) −t _(f)+τ),  (10)and after the two assumptions, τ→0 and t_(o)>>1/(t_(f)ω_(P) ²) becomesequivalent to equation (4), the lens equation: t_(o) ⁻¹+t_(i) ⁻¹=t_(f)⁻¹.

The standard deviation of the compressed electron pulse at arbitrarytime t_(a) is,

$\begin{matrix}{{\Delta\; t_{a}} = \sqrt{\frac{{t_{f}^{2}\left( {+ {4\; t_{a}^{2}\Delta\; v_{o}^{2}}} \right)} + {t_{a}^{2}} - {2\; t_{f}t_{a}}}{48\; t_{f}^{2}v_{0}^{2}},}} & (11)\end{matrix}$which is valid for an individual well. The time when the minimum pulseduration occurs is t_(a)=t_(f)

/(

+4t_(f) ²Δν_(o) ²)≈t_(f) and for experimentally realistic parameters isequal to t_(f). This implies that the thick lens does not image theinitial temporal pulse; it temporally focuses the electrons that entereach individual well. Since there is no image in the thick lens regime,the minimum temporal duration is not determined by the magnification Mas in the thin lens section, but is a given by,

$\begin{matrix}{{\Delta\; t_{f}} = {\sqrt{\frac{t_{f}^{2}{\overset{\sim}{\lambda}}^{2}\Delta\; v_{o}^{2}}{12\;{v_{o}^{2}\left( {{\overset{\sim}{\lambda}}^{2} + {4\; t_{f}^{2}\Delta\; v_{o}^{2}}} \right)}}} \cong \frac{t_{f}\Delta\; v_{o}}{v_{o}2\sqrt{3}}}} & (12)\end{matrix}$It should be noted that neither the temporal focal length nor thetemporal duration are directly dependent on the Doppler shiftedwavelength

, as long as the condition t_(o)<ν₀Δt_(o)/Δν_(o) is met.

An example illustrates what temporal foci are obtainable. A source emitselectrons with an energy distribution of 1 eV and a temporaldistribution of 100 fs. Electrons traveling at ν₀=c/3 and having anenergy E=31 keV gives a velocity distribution of Δν_(o)=1670 m/s. If thedistance between the source and the temporal lens is 10 cm, t_(o)=1.0 nsis less than ν₀Δt_(o)/Δν_(o)≈6.0 ns, satisfying the conditiont_(o)<ν₀Δt_(o)/Δν_(o) and equation (12) is then valid. If the two colorsused for the laser beams are 520 nm and 1040 nm, the Doppler-shiftedwavelength is

=740 nm. For a laser intensity of 3×10¹² Wcm⁻² (available withrepetition rates up to megahertz), the oscillation frequency in thepotential well is ω_(p)≈2×10¹² rad/s, which gives a focal time oft_(f)≈1 ps. With these parameters, equation (12) gives a temporalduration at the focus of Δt_(f)≈5 as. To support this ˜5 as electronpulse, time-energy uncertainty demands an energy spread of ˜50 eV. Theponderomotive compression imparts an energy spread to the electron pulsewhich can be estimated from ΔE˜mν₀

(2t_(f)), giving ˜50 eV similar to the uncertainty limit. This ΔE isvery small relative to the accelerating voltage in microscopy (200 keV)and only contributes to a decrease of the temporal coherence. In opticalspectroscopy such pulses can still be used as attosecond probes despitethe relatively large ΔE when the chirp is well characterized. Combiningthe anharmonicity broadening of 15 as, we conclude that ultimatelytemporal pulse durations in the attosecond regime can be reached.

In the temporal thick lens case, the use of ω and 2ω to create aco-propagating standing wave requires ν₀=c/3. However, the velocity ofthe electrons, ν₀, can be tuned by changing the angle of the two laserpulses. A co-propagating standing wave can still be obtained by forcingthe Doppler-shifted frequencies of both tilted laser pulses to be equal.A laser pulse that propagates at an angle θ with the respect to theelectron propagation direction has a Doppler-shifted frequency

=γω(1±(ν/c)cos θ), where ω is the angular frequency in the lab frame,

=ν{circumflex over (x)} is the electron velocity, and γ=1/√{square rootover (1−ν²/c²)}. When the two laser pulses are directed as shown in FIG.16, a co-propagating standing wave occurs for an electron with avelocity ν₀=c(k₁−k₂)/(k₁ cos θ₁+k₂ cos θ₂), where the laser pulsetravelling with the electron packet has a wave vector of magnitude k₁and makes an angle of θ₁ with the electron propagation axis; the secondlaser pulse traveling against has a wave vector magnitude of k₂ andangle θ₂, in the lab frame. An electron moving at ν₀ will see a standingwave with an angular frequency,

$\begin{matrix}{{= {\frac{2\left( {{\cos\;\theta_{1}} + {\cos\;\theta_{2}}} \right)}{{2\;\cos\;\theta_{1}} + {\cos\;\theta_{2}}}\gamma\;{\omega\left( {1 - \beta} \right)}}},} & (13)\end{matrix}$where 2k=k₁=2k₂ for experimental convenience, ω=kc, and the wavelengthis

=2πc/

=2π/

.

The standing wave created with arbitrary angles θ₁ and θ₂ will be tiltedwith respect to the electron propagation direction, which willtemporally smear the electron pulse. This tilting of the standing wavecan be corrected for by constraining the angles θ₁ and θ₂ to be:θ₂=arcsin(2 sin θ₁).

For θ₁=15° (forcing θ₂≈31°), electrons with velocity ν₀=0.36c (E≈33 keV)see a standing wave. A 1 eV electron energy distribution at the sourcegives a velocity distribution of Δν₀≈1630 m/s, at 33 keV. Using the samelaser intensity as in the thick lens case, and the new ν₀ and Δν_(o),the condition t_(o)<ν₀Δt_(o)/Δν_(o) is still satisfied, allowingequation (13) to be used, resulting in a duration at the focus ofΔt_(f)≈4.6 as. Using the tunable thick lens makes the experimentalrealization more practical, allowing for easy optical access andelectron energy tuning, while at the same time keeping Δt_(f)approximately the same. For additional tunability, an optical parametricamplifier can be used so that the laser pulse frequencies are notrestricted to ω and 2ω.

The ability to create electron pulses with duration from ˜10 fs to ˜10as raises a challenge regarding the measuring of their duration andshape. Two different schemes are presented here for measuring pulsescompressed by thick and thin temporal lenses. For measuring the thinlens compressed electron packet, the focused packet could be intersectedby a laser pulse with a Gaussian spatial focus as illustrated in FIG.17. An optical delay line would control the time delay between themeasuring laser pulse and the compressed electron packet. As the timedelay, Δt, is varied, so is the average energy of the electrons, asshown in FIG. 17. If the delay time is zero, then the average electronenergy will be unaffected, as there is no force. If the delay line ischanged so that the Gaussian pulse arrives early (late), then theaverage energy will decrease (increase). The change in the averageenergy is dependent on the duration of the electron pulse, and theintensity of the probing laser pulse. If the electron pulse is longerthan the duration of the measuring laser pulse, then the change in theaverage energy will be reduced. The steepness of the average energy as afunction of delay time, Ē(Δt), is a direct measure of the electron pulseduration, and using fs-pulsed electron energy loss spectra this schemecan be realized.

For the thick lens a similar method is described here. At the focalposition and time of the compressed temporal electron packet, a secondco-propagating potential is introduced. The positions of the individualwells in the second co-propagating standing wave can be moved by phaseshifting one of the two laser beams that create the probing potential(FIG. 17). By varying the phase shift, the potential slope (and hencethe force) that the electrons encounter at the focus is changed. If nophase shift is given to the probing standing wave, no average energyshift results. When a phase shift is introduced, the electrons will beaccelerated (or decelerated) by the slope of an individual well in thestanding wave, and as long as the phase stability between the electronsand the probing standing wave is appropriate, attosecond resolution canbe achieved. As the electron pulse duration becomes less than the periodof the standing wave, the average electron energy change increases. Theelectron temporal duration of the compressed electron packet can bedetermined directly by the steepness of the Ē(φ) curve.

Diffraction with focused electron probes is among the most powerfultools for the study of time-averaged nanoscale structures in condensedmatter. Embodiments of the present invention provide methods and systemsfor four-dimensional (4D) nanoscale diffraction, probing specific-sitedynamics with ten orders of magnitude improvement in time resolution, inconvergent-beam ultrafast electron microscopy (CB-UEM). Forapplications, we measured the change of diffraction intensities inlaser-heated crystalline silicon as a function of time and fluence. Thestructural dynamics (change in 7.3±3.5 ps), the temperatures (up to 366K), and the amplitudes of atomic vibrations (up to 0.084 angstroms) aredetermined for atoms strictly localized within the confined probe areaof 50-300 nm; the thickness was varied from 2 to 100 nm. A broad rangeof applications for CB-UEM and its variants are possible, especially inthe studies of single-particles and heterogeneous structures.

In fields ranging from cell biology to materials science, structures canbe imaged in real-space using electron microscopy. Atomic-scaleresolution of structures is usually available from Fourier-spacediffraction data, but this approach suffers from the averaging over theselected specimen area which is typically on the micrometer scale.Significant progress in techniques has enabled localization ofdiffraction to nanometer and even angstrom-sized areas by focusing acondensed electron beam onto the specimen. Parallel illumination with asingle electron wavevector is reshaped to a convergent beam with a spanof incident wavevectors. This method of convergent beam electrondiffraction (CBED), or electron microdiffraction, and with energyfiltering, has made possible determination of structures in 3 dimensionswith highly precise localization to areas reaching below one unit cell.The applications have been wide-ranging, from revealing bonding chargedistribution and local defects and strains in solids to detecting localatomic vibrations and correlations. Today, aberration-corrected,atomic-sized convergent electron beams enable analytical probing usingelectron-energy-loss spectroscopy (EELS) and scanning transmissionelectron microscopy (STEM).

In order to resolve structural dynamics with appropriate spatiotemporalresolution, femtosecond (fs) and picosecond (ps) electron pulses areideal probes because of their picometer wavelength and their large crosssection, resulting from the effective Coulomb interaction with atomicnuclei and core/valence electrons of matter. Typically, ultrafastelectron diffraction is achieved by initiating the physical or chemicalchange with a pulse of photons (pump) and observing the ensuing dynamicswith electron pulses (probe) at later times. By recording sequentiallydelayed diffraction frames a “movie” can be produced to reveal thetemporal evolution of the transient structures involved in the processesunder study.

FIG. 18 is a simplified schematic diagram of a CB-UEM set-up (top), andobserved low-angle diffraction discs according to an embodiment of thepresent invention. Femtosecond electron pulses are focused on thespecimen to form a nanometer-sized electron beam. Structural dynamicsare determined by initiating a change with a laser pulse and thenobserving the consequences using electron packets delayed in time.Insets (right) show the CB-UEM patterns taken along the Si [011] zoneaxis at different magnifications. At the high camera length used, onlythe ZOLZ discs indexed in the figure are visible; thekinematically-forbidden 200 disc appears as a result of dynamicscattering. In the reciprocal space representation of the diffractionprocess (bottom) the Ewald sphere has an effective thickness of 2α, theconvergence angle of the electron beam. The diamond structure of Siforbids any reflections from odd numbered Laue planes when the zone axisis [011].

Embodiments of the present invention provide CB-UEM methods and systemswith applications in the study of nanoscale, site-selected structuraldynamics initiated by ultrafast laser heating (10¹⁴ K/s). Because of thefemtosecond pulsed-electron capability, the time resolution is tenorders of magnitude improved from that of conventional TEM, which ismilliseconds; and because of beam convergence, high-angle Braggscatterings are visible with their intensities being very sensitive toboth the 3D structural changes and amplitudes of atomic vibrations. TheCB-UEM configuration is shown in FIG. 18; our chosen specimen is acrystalline silicon slab, a prototype material for such investigations.From these experiments, it is found that the structural change withinthe locally probed site occurs with a time constant of 7.3±3.5 ps, whichis on the time scale of the rise of lattice temperature known for bulksilicon. For these local sites, the temperatures measured at differentlaser fluences range from 299° K to 366° K, corresponding to vibrationalamplitude changes from 0.077 Å to 0.084 Å, respectively. The reportedresults would be impossible to obtain with conventional, parallel beamdiffraction.

The electron microscope is integrated with a fs oscillator/amplifierlaser system. The fundamental mode of the laser at 1036 nm was splitinto two beams: the first was frequency doubled to 518 nm and used toinitiate the heating of the specimen, whereas the second, which wasfrequency tripled, was directed to the microscope for extractingelectrons from the cathode. The time delay between pump and probe wasadjusted by changing the relative optical path lengths of these twopulses. The pulses were sufficiently separated in time (5 μs) to allowfor cooling of the specimen.

The electron packets were accelerated to 200 keV (corresponding to a deBroglie wavevector of 39.9 Å⁻¹), de-magnified, and finally focused (witha 6 mrad convergence angle) to an area of 50-300 nm diameter on thewedge-shaped specimen, as shown in FIG. 18. A wide range of thicknesses,starting from ˜2 nm was accessible simply by moving the electron beamlaterally. The silicon specimen was prepared by mechanical polishing ofa wafer along the (011) planes, followed by Ar ion-milling for finalthinning/smoothing; the wedge angle was 2°. In the microscope, Kikuchilines were observed and used as a guide to orient the specimen with the[011] zone axis either parallel or tilted relative to the incidentelectron beam direction.

FIG. 18 display the typical high-magnification (high-value cameralength) CB-UEM patterns of Si obtained when the specimen is unexcitedand the zone axis is very close to [011]; the magnification (>10×) cabbe seen by comparing the disc length scale in FIG. 18 and ring radius inFIG. 19. Unlike parallel-beam diffraction which yields spots,convergent-beam diffraction produces discs in reciprocal space (backfocal plane of the objective lens) with their diameter given by theconvergence angle (2α) of the electron pulses. These discs form the ZeroOrder Laue Zone (ZOLZ) of the pattern; they show white contrast withthin specimens and exhibit the interference patterns displayed in FIG.18 when the thickness is increased.

In the reciprocal space, the effective thickness of the Ewald sphere is2α (bottom panel of FIG. 18), giving rise to multiple spheres that canintersect with Higher Order Laue Zones (HOLZ) reflections, the focus ofthis study (see FIG. 19) and the key to 3D structural information; thefirst and second zones, FOLZ and SOLZ, are examples of such zones orrings. The interference patterns in the disks are the result ofdynamical scattering in silicon and are reproduced in our CB-UEMpatterns (FIG. 18).

The scattering vectors of HOLZ rings (R) are related to the inter-zonespacing in the reciprocal space (h_(z) in Å⁻¹) by the tilt angle fromthe zone axis (η) and by the magnitude of the incident electron'swavevector (k₀). In the plane of the detector and for our tilt geometry,the HOLZ ring scattering vector is given by (equation (14)):R≅(k ₀ ² sin²(η)+2k ₀ h _(z))^(1/2) −k ₀ sin(η),  (14)where, for our case of the [011] zone axis, h_(z)=n/(a√{square root over(2)}) with n=1, 2, 3 . . . for the different Laue zones. Additionally,for this zone axis, k+l=n, where (hkl) are the Miller indices of thereciprocal space. When k+l=1, for FOLZ, k and l must have differentparity, which is forbidden by the symmetry of the diamond Si structure.Therefore, the FOLZ along the [011] zone axis should be absent and thefirst visible ring should belong to SOLZ; in general, all odd numberedzones will be forbidden. Here, HOLZ indexing is defined according to thefcc unit cell and not to the primitive one [1].

FIG. 19 illustrates temporal frames obtained using CBUED. In FIG. 19( a)high angle SOLZ ring obtained for a tilt angle of 5.15° from the [011]zone axis are shown. Besides SOLZ, Kikuchi lines and periodic bands (dueto atomic correlations) are visible. The ZOLZ discs are blocked (topleft) to enhance the dynamic range in the area of interest; the disc ofthe direct beam (the center one in FIG. 18 discs) is indicated by acircle. The intensity scale is logarithmic. In FIG. 19( b,c,d) timeframes of the SOLZ ring are shown by color mapping for visualization ofdynamics. The intensity of the ring changes within picoseconds, but thesurrounding background remains at the same level.

FIG. 19( a) presents the HOLZ ring taken with the CB-UEM. In order toreduce the strong on-zone-axis dynamic scattering (and to bring the highscattering angles into the range of the recording camera), the slab wastilted 5.15° away from the [011] zone axis, along the [02 2] direction.The scattering vector of the Bragg points of the ring, from the directbeam position, was measured to be 2.2 Å⁻¹, close to the value of 2.22Å⁻¹ obtained by using equation (14) for n=2, which identifies the spotsshown as part of the SOLZ. From this value, the know lattice separationof 5.4 Å was obtained for silicon.

In addition to the SOLZ ring, Kikuchi lines and some oscillatory bandsare also visible in the CB-UEM, as seen in FIG. 19( a). Kikuchi linesarise from elastic scatterings of the inelastically scattered electrons,whereas the oscillatory bands in the thermal diffuse scattering (TDS)background result from correlations between the atoms. We also observeddeficit HOLZ lines and interference fringes in ZOLZ discs for a two-beamcondition.

The temporal behavior is displayed in FIG. 19, with three CB-UEM framestaken at time delays of t=−14.8 ps, +5.2 ps, and +38.2 ps, together witha static image; the zero of time is defined by the coincidence of thepump and probe pulses in space and time. The frame at negative time hashigher ring intensity than that observed at +38.2 ps, whereas the +5.2ps frame shows an intermediate intensity value. It is clear from theresults that the intensity change is visible within the first 5 ps ofthe structural dynamics. For quantification, the intensities in eachframe were normalized to the area of azimuthally integrated background.The normalization of the HOLZ ring intensities to the TDS backgroundmakes the atomic vibration estimations insensitive to the thicknesschanges of the probed area, which may result from slight beam jittering.

FIG. 20 illustrates diffraction intensities at different times andfluences. Normalized, azimuthally-integrated intensity changes of theSOLZ ring are shown with time ranging from −20 ps to +100 ps, for twodifferent laser powers. Whereas the 10 mW response does not shownoticeable dynamics, the 107 mW transient has a clear intensity changewith a characteristic time of 7.3±3.5 ps. The range of fluences studiedwas 1.7 to 21 mJ/cm² (see FIG. 21). The red curve is a mono-exponentialfit based on the Debye-Waller effect. The red dashed line through the 10mW data is an average of the points after +20 ps. The dependence onfluence is given in FIG. 21.

FIG. 20 depicts the transient behavior of the SOLZ ring intensity fortwo different laser power, 10 mW and 107 mW, corresponding to pulsefluence of 1.7 and 19 mJ/cm², respectively; the heating laser beamdiameter on the specimen is 60 μm. The intensities were normalized tothe average value obtained at negative times. Whereas the intensitychange is essentially absent in the 10 mW data, the results for the 107mW set shows a transient behavior with a characteristic time of 7.3±3.5ps, obtained from the mono-exponential fit shown in red in the figure.The temporal response of UEM-2 is on the fs time scale, as obtained byEELS, and it is much shorter than the 7 ps illustrated here.

The local heating of the lattice is responsible for the SOLZ intensitychange with time. A pump laser, in our case at 518 nm (2.4 eV), excitesthe valance electrons of Si to the conduction band; one-photonabsorption occurs through the indirect bandgap at 1.1 eV, andmulti-photon absorption excites electron-hole pairs through the directgap. The excited carriers thermalize within 100 fs, via carrier-carrierscatterings, and then electron cooling takes place in ˜1 ps, byelectron-phonon coupling. During this time lattice heating occursthrough increased atomic vibration, reducing SOLZ intensity. Theeffective lattice temperature is ultimately established with a timeconstant of a few picoseconds depending on density of carriers orfluence. However, in CB-UEM measurements the lattice-temperature risecould be slower than in bulk depending on the dimension of the specimenrelative to the mean free path of electrons in the solid.

The dynamical change can be quantified by considering a time-dependentDebye-Waller factor with an effective temperature describing thedecrease in the Bragg spot intensity with time. If the root-mean-square(rms) displacement of the atoms,

u_(x) ²

^(1/2), along one of the three principle axes is denoted by u_(x) forsimplicity, and the scattering vector by s, then the HOLZ ring intensitycan be expressed as (equation (15)):I _(Ring) ^(F)(t)I ₀(t ⁻)exp[−4π² s ² u _(x) ²(t)],  (15)where I_(Ring) ^(F)(t) is the measured intensity for a given fluence, F,and the vibrational amplitude is now time dependent. Note that u_(x) is⅓ of the total, u_(total).

In the Einstein model of atomic vibrations, which has been usedsuccessfully for silicon, the atoms are treated as independent harmonicoscillators, with the three orthogonal components of the vibrationsdecoupled. As a result, a single frequency (ω) is sufficient to specifythe energy eigenstates of the oscillators. The relationship of thevibrational amplitude to temperature can be established by simplyconsidering the Boltzmann average over the populated eigenstates.Consequently, the probability distribution of atomic displacements isderived to be of Gaussian form, with a standard deviation correspondingto the rms (u_(x)) of the vibration involved (equation (16)):u _(x)=[(h/2ωm)coth(hω/2k _(B) T _(eff))]^(1/2)  (16)where h is Planck's constant, k_(B) the Boltzmann constant, T_(eff) inour case the effective temperature, and m the mass of the oscillator. Inthe high temperature limit, i.e. when hω/2k_(B)T<<1, eq. 3 simplifies tomω²u_(x) ²=k_(B)T, which is the classical limit for a harmonicoscillator; the zero-point energy, which contributes almost half of themean vibration amplitude at room temperature, is included in equation(16). The value of hω is 25.3 meV. Despite its simplicity, the Einsteinmodel in equation (16) was remarkably successful in predicting the HOLZrings and TDS intensities by multi-slice simulations.

FIG. 21 illustrates the amplitudes of atomic vibrations (rms) plottedagainst the observed intensity change at different fluences. The insetshows the mono-exponential temporal behavior, with the asymptoteshighlighted (circles) for their values at different fluences. Thefluence was varied from 1.7 to 21 mJ/cm². This comparative study of theeffect of the fluence was performed at a slightly different sample tilt(corresponding to s=2.7 Å⁻¹), corresponding to a thickness of ˜80 nm.For each fluence, the temperature represents the effective value for thelattice structural change. The error bars given were obtained from thefits at the asymptotes shown in the inset, and they are determined bythe noise level of temporal scans.

In FIG. 21, we present the change in the asymptotic intensity withfluence (inset), and the derived vibrational amplitudes for thedifferent temperatures. The amplitudes are directly obtained fromequation (15), as s is experimentally measured. The relative temperaturechange (from t⁻ to t₊) is then derived from equation (16), taking thevalue of u_(x) at room temperature (297° K) to be 0.076 Å. The amplitudeof atomic vibrations, and hence the temperature, increases as thefluence of the initiating pulse increases. Although the trend isexpected for an increased u_(x) with temperature, the absolute values,from 0.077 to 0.084 Å, correspond to a large 3.2% to 3.6% change innearest neighbor separation; these values are still well below the 15%criterion for a melting phase transition.

The linear thermal expansion coefficient has been accurately determinedfor silicon, and for a value of 2.6×10⁻⁶K⁻¹ at room temperature thevibrational amplitudes reported here are much higher than theequilibrium thermal values at the same temperature. This is because theeffective temperature applies to a lattice arrested in a picosecond timewindow; at longer times, the vibrations equilibrate to a lowertemperature. As such, measuring nanoscale local temperatures on theultrashort time scale enhances the sensitivity of the probe thermometerby orders of magnitude. Moreover, the excitation per site issignificantly enhanced. For a single-photon absorption at the fluenceused, we estimate, for a 60 nm-thick specimen, the number of absorbedphotons per Si atom (for the fs pulse employed) to be ˜0.01, as opposedto 10⁻⁹ photons per atom if the experiments were conducted in thetime-averaged mode.

The achievement of nanoscale diffraction with convergent-beam ultrafastelectron microscopy opens the door to exploration of differentstructural, morphological, and electronic phenomena. The spatiallyfocused and timed electron packets enable studies of single particlesand structures of heterogeneous media. Extending the methodologyreported here to other variants, such as EELS, STEM and nanotomography,promises possibilities for mapping individual unit cells and atoms onthe ultrashort time scale of structural dynamics.

With 4D electron microscopy, in situ imaging of the mechanical drummingof a nanoscale material is measured. The single crystal graphite film isfound to exhibit global resonance motion that is fully reversible andfollows the same evolution after each initiating stress pulse. At earlytimes, the motion appears “chaotic” showing the different mechanicalmodes present over the micron scale. At longer time, the motion of thethin film collapses into a well defined fundamental frequency of 0.54MHz, a behavior reminiscent of mode locking; the mechanical motion dampsout after ˜200 μs and the oscillation has a “cavity” quality factor of150. The resonance time is determined by the stiffness of the materialand for the 53-nm thick and 55-μm wide specimen used here we determinedYoung's modulus to be 0.8 TPa, for the in-plane stress-strain profile.Because of its real-time dimension, this 4D microscopy has applicationsin the study of these and other types of materials structures.

Structural, morphological, and mechanical properties of materials havedifferent length and time scales. The elementary structural dynamics,which involve atomic movements, are typically of picometer length scaleand occur on the time scale of femto (fs) to picoseconds (ps).Collective phenomena of such atomic motions, which define morphologicalchanges, are observed on somewhat longer time scale, spanning the ps tonanosecond (ns) time domain, and the length scale encompasses up tosub-micrometers. These microscopic structures are very different inbehavior from those involved in the mechanical properties. On thenanoscale, when the membrane-like mechanical properties have highfrequencies and complex spatial-mode structures, imaging becomes ofgreat value in displaying the spatiotemporal behavior of the materialunder stress.

Utilizing embodiments of the present invention, we have visualizednanoscale vibrations of mechanical drumming in a single-crystallinegraphite film (53-nm thick). To study the transient structures, in bothspace and time, our method of choice has been 4D ultrafast electronmicroscopy (UEM). This microscope enables investigation of the atomicstructural and morphological changes in graphite on the fs to ns timescale and for nm-scale resolution. Additionally, mechanical propertiescan be determined in real time, which are evident on the ns andmicrosecond (μs) time scale. The stress is introduced impulsively usinga ns laser pulse while observing the motions in real space (in situ) inthe microscope using the stroboscopic electron pulses. Remarkably, attimes immediately following the initiating pulse the motion appears“chaotic” in the full image transients, showing the different mechanicalmodes present in graphite. However, after several μs the motion of thenanofilm collapses into a final global resonance of 0.54 MHz. From thisresonance of mechanical drumming of the whole plate, we obtained thein-plane Young's modulus of 0.8 terapascal (Tpa). The reported coherentresonance represents the in-phase build up of a mechanical drumming,which is directly imaged without invasive probes.

Graphite was chosen because of its unique material properties; it ismade of stacked layers of 2D graphene sheets, in which the atoms of eachsheet are covalently bonded in a honeycomb lattice, and the sheetsseparated by 0.335 nm are weakly held together by van der Waals forces.It displays anisotropic electromechanical properties of high strength,stiffness, and thermal/electric conductivity along the 2D basal planes.More recently, with the rise of graphene, a new type ofnano-electromechanical system (NEMS) has been highlighted with aprototypical NEMS being a nanoscale resonator, a beam of material thatvibrates in response to an applied external force. With the thicknessesreaching the one atomic layer, graphene remains in a high crystallineorder, resulting in a NEMS with extraordinary thinness, large surfacearea, low mass density, and high Young's modulus.

Briefly, the setup for ultrafast (and fast) electron imaging involvesthe integration of laser optical systems into a modified transmissionelectron microscope (TEM). Upon the initiation of a structural change byeither heating of the specimen or through electronic excitation by thelaser pulses, an electron pulse generated by the photoelectric effect isused to probe the specimen with a well-defined time delay. A microscopyimage or a diffraction pattern is then taken. A series of time-framedsnapshots of the image or the diffraction pattern recorded at a numberof delay times provides a movie, which displays the temporal evolutionof the structural (morphological) and mechanical motions, using eitherthe fs or ns laser system.

Because here the visualization is that of the mechanical modes withresonances on the MHz scale, the ns resolution was sufficient. Theelectrons are accelerated to 200 kV with a de Broglie wavelength of2.5079 pm. Two laser pulses were used to generate the clocking,excitation pulse at 532 nm and another at 355 nm for the generation ofthe electron pulse for imaging. The time delay was controlled bychanging the trigger time for electron pulses with respect to that ofclocking pulses. The delay can be made arbitrarily long and therepetition rate varies from a single shot to 200 kHz, to allow completeheat dissipation in the specimen. The experiments were carried out witha natural single crystal of graphite flakes on a TEM grid. Graphiteflakes were left on the surface, covering some of the grid squarescompletely. The observed dynamics are fully reversible, retracing theidentical evolution after each initiating pulse; each image isconstructed stroboscopically, in a half second, from typically 2500pulses of electrons and completing all time-frames (movies) in twentyminutes.

FIG. 22 illustrates images and the diffraction pattern of graphite. (A),an image shows features of fringes in contrast (scale bar: 5 μm). Samplethickness was measured to be 53 nm using electron energy lossspectroscopy (EELS). (B) Magnified view of the indicated square of panelA (scale bar: 1 μm). (C) Diffraction pattern obtained by using aselected area diffraction aperture (SAD), which covered an area of 6 μmin diameter on the specimen. The incident electron beam is parallel tothe [001] zone axis. Bragg spots are indexed as indicated for somerepresentative ones.

Panels A and B of FIG. 22 show the UEM (bright field) images ofgraphite, and in panel C, a typical electron diffraction pattern isgiven. The Bragg spots are indexed according to the hexagonal structureof graphite along the [001] zone axis, with the lattice dimension ofa=b=2.46 Å (c=6.71 Å). In FIG. 22A, and at higher magnification in FIG.22B, contrast fringes are clear, typically consisting of linear fringeshaving ˜1 μm length and a few hundred-nm spacing. These contrast fringesare the result of physical bucking of the graphene layers by constraintsor by nanoscale defects within the film. In the dark regions, the zoneaxis (the crystal [001]) is well aligned with the incident electron beamand electrons are scattered efficiently, whereas in the lighter regionsthe alignment of the zone axis deviates more and the scatteringefficiency is lower. With these contrast patterns, changes in imageprovide a sensitive visual indicator of the occurrence of mechanicalmotions. The black spots are natural graphite particles.

FIG. 23 illustrates representative image snapshots and differenceframes. (A) Images recorded stroboscopically at different time delays,indicated at the top right corner of each image (t₁, t₂, t₃, t₄, andt₅), after heating with the initiating pulse (fluence=7 mJ/cm²); t₁=200ns; t₂=500 ns; t₃=10 μs; t₄=30 μs; t₅=60 μs; and the negative time framewas taken at −1000 ns. Note the change in position of fringes with time,an effect that can be clearly seen in FIG. 23B. (B) Image differenceframes with respect to the image taken at −1 μs, i.e., Im(−1 μs; t),which show the image change with time. The reversal in contrast clearlydisplays the oscillatory (resonance) behavior.

In FIG. 23(A), we display several time-framed images of graphite takenat a repetition rate of 5 kHz and at delay times indicated with respectto the clocking (heating) pulse with the fluence of 7 mJ/cm². Atpositive times, following t=0, visual changes are seen in the contrastfringes. With time, the contrast fringes change their location in theimages, and with these and other micrographs of equal time steps we madea movie of the mechanical motions of graphite following the nsexcitation impulse. To more clearly display the temporal evolution onthe nanoscale, image-difference frames were constructed.

In FIG. 23(B), depicted are the images obtained when referencing to the−1 μs frame, i.e., Im(−1 μs; t). In the difference images, the regionsof white or black indicate locations of surface morphology change(contrast pattern movement), while gray regions are areas where thecontrast is unchanged from that of the reference frame. Care was takento insure the absence of long-term specimen drifts as they can causeapparent contrast change; note that in the difference images, the staticfeatures are not present. The image changes, reported in this study, arefully reproducible, retracing the identical evolution after eachinitiating laser pulse, as mentioned above. The reversal of contrastwith time in FIG. 23(B) directly images the oscillatory behavior of thedrumming.

The image change was quantified by using the method ofcross-correlation. The normalized cross correlation of an image at timet with respect to that at time t′ is expressed as

$\begin{matrix}{{\gamma(t)} = \frac{\sum\limits_{x,y}^{\;}\;{{C_{x,y}(t)}{C_{x,y}\left( t^{\prime} \right)}}}{\sqrt{\sum\limits_{x,y}^{\;}\;{{C_{x,y}(t)}^{2}{\sum\limits_{x,y}^{\;}\;{C_{x,y}\left( t^{\prime} \right)}^{2}}}}}} & (17)\end{matrix}$where the contrast C_(x,y)(t) is given by [I_(x,y)(t)−Ī(t)]/Ī(t), andI_(x,y)(t) and I_(x,y)(t′) are the intensities of pixels at the positionof (x,y) at times t and t′; Ī(t) and Ī(t′) are the means of I_(x,y)(t)and I_(x,y)(t′), respectively. This correlation coefficient γ(t) is ameasure of the temporal change in “relief pattern” between the twoimages being compared, which can be used as a guide to image dynamics asa function of time. Shown in FIG. 24 are cross-correlation valuesbetween the image at each measured time point and a reference imagerecorded before the arrival of the clocking pulse.

FIG. 24 illustrates the time dependence of image cross correlation. Thewhole scan for 100 μs is made of 2000 images taken at 50-ns steps. Alsodepicted are the zoomed-in image cross-correlations of threerepresentative time regimes (I, II, and III). In each zoomed-in panel,the selected-area image dynamics of five different regions are included.Note the evolution from the “chaotic” to the global resonance (drumming)behavior at long times.

Over all pixels, the time scale for image change covers the full rangeof time delays, from tens of ns to hundreds of μs, indicating thecollective averaging over the sites of the specimen. Upon impulsiveheating at t=0, the image cross-correlation changes considerably with anappearance of a “chaotic” behavior, in the ˜5 μs range (regime I in FIG.24). After 10 μs, e.g., regime II, the cross correlation change beginsto exhibit periodicity (regime II), and at longer time, a well-definedresonance oscillation emerges (regime III). This is also evident in theselected-area image dynamics (SAID) in several regions (noted as 1 to 5)where the temporal behavior is of different shapes at early time butconverges into a single resonance transient after several tens of μs.The shape of image cross correlation dynamics was robust at differentfluences, from 2 to ˜10 mJ/cm², but the amplitude varies.

The overall decay of the transients is on a time scale shorter than theseparation between pulses. In fact, we have verified the influence ofrepetition rate and could establish the full recovery at the timeintervals indicated. Heat transfer must occur laterally. With an initialz-independent heat profile by absorption of the heating pulse ingraphite, we estimated, using a 2D heat diffusion in a homogeneousmedium, the time scale for an in-plane transfer, with thermalconductivity λ=5300 W/(m·K), density ρ=2260 kg/m³, and specific heatc_(V)=707 J/(K·kg). For the radius at half height of the initial pulseheat distribution r₀=30 μm, t_(1/2), the time for the axial temperatureto drop to a half of its initial value, is deduced to be ˜720 ns,certainly much shorter than the 200-μs time interval between pulses. Itfollows that the decay of the oscillation [Q/(π·f₀)], as derived below,is determined by the damping of mechanical motions.

When the specimen absorbs intense laser light, the lattice energy,converted from carriers (electron energy) by electron-phonon coupling,in a few ps, builds up in the illuminated spot on the surface within theduration of the laser pulse. As a consequence, the irradiated volumewill expand rapidly following phonon-phonon interaction on the timescale of tens of ps. The resulting thermal stress can induce mechanicalvibration in the material, but a coherent oscillatory behavior, due tothe thermoelastic stress, will only emerge in the image if the impulsivestress is on a time scale shorter than the period; probing of imagesshould be over the entire time scale of the process, in this case 100μs. On the ultrashort time scale we have observed the structural andmorphological elastic changes.

FIG. 25 illustrates resonance dynamics and FFT of graphite. (Left) Timedependences of image cross correlation of full image (A) and imageintensity on the selected area of 4×6 pixels as indicated by thearrowhead (B) in FIG. 24. (Right) Fast Fourier transforms of imagecross-correlation (C: 0-100 μs; D: 60-100 μs) and image intensity (E:0-100 μs; F: 60-100 μs). Asterisks in the panels indicate overtones.Note the emergence of the resonance near 1 MHz in panel F.

The resonance modes in graphite are highlighted in FIG. 25 by taking thefast Fourier transform (FFT) of image cross-correlation in the timeregime of 0-100 μs. The FFT (FIG. 25C) shows several peaks of differentfrequencies, among which the strongest one around 2.13 MHz is attributedto the overtone of 1.08 MHz. The overtones, due to the truncated natureof cross-correlation close to the value of 1, are greatly reduced in theFFT of image intensity change (FIGS. 25E and 25F). In a few tens of μs,various local mechanical modes observed at early time damp out and oneglobal mode around 1 MHz survives. The peak when fitted to a Lorentzianyields a resonant frequency of 1.08 MHz, and a “cavity” quality factor Q(=f₀/Δf)=150±30. This dominant peak gives the fundamental vibration modeof the plate in graphite. For a period of vibration, the contrastpattern of image would recur twice to its initial feature giving theobserved frequency to be twice that of structural vibration; thefundamental frequency is, thus, obtained to be 0.54 MHz.

A square mechanical resonator clamped at four edges without tension hasa fundamental resonance mode of f₀ which is given by

$\begin{matrix}{f_{0} = {{A{\frac{d}{L^{2}}\left\lbrack \frac{Y}{\left( {1 - v^{2}} \right)\rho} \right)}^{1\text{/}2}} + {f(T)}}} & (18)\end{matrix}$where f(T) due to tension T is zero in this case. Y is the Young'smodulus; ρ is the mass density; ν is the Poisson's ratio; L is thedimension of a grid square; d is the thickness of the graphite; and A isa constant, for this case equal to 1.655. We measured d to be 53 nm fromEELS. Knowing ρ=2260 kg/m³ (300 K), ν=0.16 for graphite, and L=55 μm, weobtained from the observed resonance frequency the Young's modulus to be0.8 TPa, which is in good agreement with the in-plane value of 0.92 TPa,obtained using stress-strain measurements. This value is different bymore than an order of magnitude from the c-axis value we measured usingthe microscope in the ultrafast mode of operation.

Thus, using embodiments of the present invention, we have demonstrated avery sensitive 4D microscopy method for the study of nanoscalemechanical motions in space and time. With selected-area-imagingdynamics, the evolution of multimode oscillations to a coherentresonance (global) mode at long time provides the mapping of localregions in the image and on the nanoscale. The time scale of theresonance is directly related to materials anisotropic elasticity(Young's modulus), density, and tension, and as such the reportedreal-time observation in imaging can be extended to study mechanicalproperties of membranes (graphene in the present case) and othernanostructures with noninvasive probing. The emergent propertiesresolved here are of special interest to us as they represent awell-defined “self-organization” in complex macroscopic systems.

The function of many nano and microscale systems is revealed when theyare visualized in both space and time. Here, four-dimensional (4D)electron microscopy provided in accordance with an embodiment of thepresent invention is used to measure nanomechanical motions ofcantilevers. From the observed oscillations of nanometer displacementsas a function of time, for free-standing beams, we are able to measurethe frequency of modes of motion, and determine Young's elastic modulus,the force and energy stored during the optomechanical expansions. Themotion of the cantilever is triggered by molecular charge redistributionas the material, single-crystal organic semiconductor, switches from theequilibrium to the expanded structure. For these material structures,the expansion is colossal, typically reaching the micron scale, themodulus is 2 GPa, the force is 600 μN, and the energy is 200 pJ. Thesevalues translate to a large optomechanical efficiency (minimum of 1% andup to 10% or more), and a pressure of nearly 1,500 atm. We note that theobservables here are real-material changes in time, in contrast to thosebased on changes of optical/contrast intensity or diffraction.

As the physical dimensions of a structure approach the coherence lengthof carriers, phenomena not observed on the macroscopic scale (e.g.,quantization of transport properties) become apparent. The discovery andunderstanding of these quantization effects requires continued advancesin methods of fabrication of atomic-scale structures and, asimportantly, in the determination of their structural dynamics inreal-time when stimulated into a configuration of a nonequilibriumstate. Of particular importance are techniques that are noninvasive andcapable of nanoscale visualization in real-time.

Examples of the rapid progress in the study of nanoscale structures arenumerous in the field of micro and nanoelectromechanical systems (i.e.,MEMS and NEMS, respectively). Recent advancements have resulted instructures having single-atom mass detection limits and bindingspecificities on the molecular level, and especially for biologicalsystems. Beyond mass measurement and analyte detection, changes in thedynamics of these nanoscale structures have been shown to be sensitiveto very weak external fields, including electron and nuclear spins,electron charge, and electron and ion magnetization. The response toexternal stimuli is manifested in deflections of the nanoscale, and avariety of techniques have been used to both actuate and detect thesmall-amplitude deflections. Optical interference is often used formeasurement purposes, wherein the deflections of the structure cause aphase shift in the path-stabilized laser light thus providing detectionsensitivities that are much less than the radius of a hydrogen atom.

High spatiotemporal resolutions (atomic-scale) can be achieved in 4Dultrafast electron microscopy (UEM). Thus it is possible to imagestructures, morphologies, as well as nanomechanical motions (e.g.,nanogating and nanodrumming) in real-time. Using embodiments of thepresent invention, we direct visualized nano and microscale cantilevers,and the (resonance) oscillations of their mechanical motions. The staticimages were constructed from a tomographic tilt series of images,whereas the in situ temporal evolution was determined using thestroboscopic configuration of UEM, which is comprised of an initiating(clocking) laser pulse and a precisely-timed packet of electrons forimaging. The pseudo-one-dimensional molecular material (copper7,7,8,8-tetracyanoquinodimethane, [Cu(TCNQ)]), which forms singlecrystals of nanometer and micrometer length scale, is used as aprototype. The optomechanical motions are triggered by charge transferfrom the TCNQ radical anion (TCNQ⁻) to copper (Cu⁺). More than athousand frames were recorded to provide a movie of the 3D movements ofcantilevers in time. As shown below, the expansions are colossal,reaching the micrometer scale, and the spatial modes are resolved on thenanoscale in the images (and angstrom-scale in diffraction) withresonances of megahertz frequencies for the fixed-free cantilevers. Fromthese results, we obtained the Young's modulus, and force and energystored in the cantilevers.

Here, different crystals were studied and generally are of two types:(1) those “standing”, which are free at one end (cantilevers), and (2)those which are “sleeping” on the substrate bed; the latter will be thesubject of another report. For cantilevers, the dimensions of the twocrystals studied are 300 nm thick by 4.6 μm long and 2.0 μm thick by 10μm long (see FIG. 26). As such, they define an Euler-Bernoulli beam, forwhich we expect the fundamental flexural modes to be prominent, besidesthe longitudinal one(s) which are parallel to the long axis of thecrystal.

Our interest in Cu(TCNQ) stems from its highly anisotropic electricaland optical properties, which arise from the nature of molecularstacking in the structure. As illustrated in FIG. 26, Cu(TCNQ) consistsof an interpenetrating network of discrete columns of Cu⁺ and TCNQ⁻running parallel to the crystallographic a-axis. The TCNQ moleculesorganize so that the π-systems of the benzoid rings are stronglyoverlapped, and the favorable interaction between stacked TCNQ moleculesmakes the spacing between the benzoid rings only 3.24 Å, significantlyless than that expected from purely van der Waals-type interactions. Itis this strong π-stacking that results in the pseudo-one-dimensionalmacroscale crystal structure and is responsible for the anisotropicproperties of the material. With electric field or light, the materialbecomes mixed in valence with both Cu⁺(TCNQ⁻) and Cu^(o) (TCNQ^(o)) inthe stacks, weakening the interactions and causing the expansion. Athigh fluences, the reversible structural changes become irreversible dueto the reduction of copper from the +1 oxidation state to copper metaland subsequent formation of discrete islands of copper metal driven byOstwald ripening. The methodology we used here for synthesis resulted inthe production of single crystals of phase I.

FIG. 26 illustrates atomic to macro-scale structure of phase I Cu(TCNQ).Shown in the upper panel is the crystal structure as viewed along thea-axis (i.e., π-stacking axis) and c-axis. The unit cell is essentiallytetragonal (cf. ref. 19) with dimensions: a=3.8878 Å, b=c=11.266 Å,α=γ=90°, β=90.00(3)°; gray corresponds to carbon, blue corresponds tonitrogen, and yellow corresponds to copper. The hydrogen atoms on thesix-membered rings are not shown for clarity. The lower panel displays atypical selected-area diffraction pattern from Cu(TCNQ) single crystalsas viewed down the [011] zone axis along with a micrograph taken in ourUEM. The rod-like crystal habit characteristic of phase I Cu(TCNQ) isclearly visible.

FIG. 27 illustrates a tomographic tilt series of images. The frames showimages (i.e., 2D-projections) of the Cu(TCNQ) single crystals acquiredat different tilt angles of the specimen substrate. The highlightedregion illustrates a large change in the position of the free-standingmicroscale crystal relative to another, which is lying flat on thesubstrate, as we change the tilt angle. The scale bar in the lower leftcorner measures two micrometers. The tilt angle at which each image wasacquired is shown in the lower right corner of each frame in degrees.The tilt angle is defined as zero when the specimen substrate is normalto the direction of electron propagation in the UEM column.

The tilt series images shown in FIG. 27 provide the 3D coordinates ofthe cantilevers. The dimensions and protrusion angles of thesefree-standing crystals were characterized by taking static frames atdifferent rotational angles of the substrate. By placing the crystalprojections into a laboratory frame orthogonal basis and measuring thelength of the projections in the x-y (substrate) plane as the crystal isrotated by an angle α about the x-axis, the measured projections wereobtained to be Θ of 37.8° and φ of 25.3°, where Θ is the angle thematerial beam makes with respect to substrate-surface normal and φ isthe azimuthal angle with respect to the tilt axis, respectively. Notethat the movie of the tilt series clearly shows the anchor point of thecrystal to be the substrate. The dimensions and geometries of thecrystals are determined from the tilt series images with 5% precision.

To visualize real-time and space motions, the microscope was operated at120 kV and the electron pulses were photoelectrically generated by laserlight of 355 nm. The clocking optical pulses (671 nm laser), which arewell-suited to induce the charge transfer in Cu(TCNQ), were heldconstant at 3 μJ, giving a maximum fluence of 160 mJ/cm². Because therelevant resonance frequencies are on the MHz scale, the ns pulsearrangement of the UEM was more than enough for resolving the temporalchanges. The time delay between the initiating laser pulse and probeelectron pulse was controlled with precision, and the repetition rate of100 Hz ensured recovery of the structure between pulses. A typicalstatic image and selected-area diffraction are displayed in FIG. 26.From the selected-area diffraction and macroscopic expansion we couldestablish the nature of correlation between unit cell and the crystalchange.

The 4D space-time evolution of cantilevers is shown in FIGS. 28 and 29.The referenced (to negative time, t_(ref)=−10 ns; i.e., before thearrival of the clocking pulse) difference images of the microscale (FIG.28) and nanoscale (FIG. 29) free-standing single crystal clearly displaymodes of expansion on the MHz scale. Each image illustrates how thespatial location of the crystal has changed relative to the referenceimage as a function of the time delay, elucidating both the longitudinaland transverse displacements from the at-rest position. In order toaccurately measure the positions in space we used a reference particlein the image. These reference particles, which are fixed to the surfaceof the substrate, do not appear in frame-reference images if drift isabsent or corrected for. This is an important indication that theobserved crystal dynamics do not arise from motion of the substrate dueto thermal drift or photothermal effects. Moreover, there is nosignificant movement observed in images obtained before the arrival ofthe excitation pulse, indicating that, during the time of pulseseparation, the motion has completely damped out and the crystal hasreturned to its original spatial configuration. The thermal, charging,and radiation effects of the electron pulses are negligible here and inour previous studies made at higher doses. This is evidenced in the lackof blurring of the images or diffraction patterns; no beam deflectiondue to sample charging was observed. Lastly, no signs of structuralfatigue or plasticity were observed during the course of observation,showing the function of the cantilever to be robust for at least 10⁷pulse cycles.

Shown in FIG. 30 is the displacement of the microscale single crystal asa function of time, in both the longitudinal and transverse directions,along with the fast Fourier transforms (FFT) of the observed spatialoscillations for the time range shown (i.e., 0 to 3.3 μs). The motionsin both directions of measurement are characterized by a large initialdisplacement from the at-rest position. The scale of expansion isenormous. The maximum longitudinal expansion possible (after accountingfor the protrusion angle) for the 10 μm crystal would be 720 nm or over7% of the total length. For comparison, a piezoelectric material such aslead zirconate titanate has typical displacements of less than 1% fromthe relaxed position, but it is known that molecular materials can showenormous optically-induced elastic structural changes on the order of10% or more. The large initial motion is transferred into flexural modesin the z and x-y directions, and these modes persist over themicrosecond (or longer) scale. The overall relaxation of the crystal toits initial position is not complete until several milliseconds afterexcitation. From the FFTs of the measured displacements, we obtained thefrequency of longitudinal oscillation to be 3.3 MHz, whereas thetransverse oscillations are found at 2.5 and 3.3 MHz (FIG. 30). We notethat the motion represents coupling of modes with dephasing, so it isnot surprising that the FFT gives more than one frequency. In fact, froman analysis consisting of a decomposition of the motion via rotation ofa principle axes coordinate system relative to the laboratory frame, wefound that the plane of lateral oscillation of the crystal was tilted by18° relative to the plane of the substrate. The nature of contact withthe substrate influences not only the mode structure but also thedamping of cantilevers.

Because of the boundary conditions of a fixed-free beam, the vibrationnodes are not evenly spaced and the overtones are not simple integermultiples of the fundamental flexural frequency (f₁), but rather occurat 6.26, 17.5, and 34.4 for f₂, f₃, and f₄, respectively. This is instark contrast to the integer multiples of the fundamental frequency ofa fixed-fixed beam. Taking 3 MHz to be the main fundamental flexuralfrequency of the microscale crystal, we can deduce Young's elasticmodulus of the crystal. The expression for the frequencies of transverse(flexural) vibrations of a fixed-free beam is given by,

$\begin{matrix}{f_{n} = {{\eta\frac{\pi\;\kappa}{8\; L^{2}}c} \equiv {\eta\frac{\pi\;\kappa}{8\; L^{2}}\sqrt{\frac{Y}{\rho}}}}} & (19)\end{matrix}$where f_(n) is the frequency of the n^(th) mode in Hz, L is the beamlength at rest, Y is Young's modulus, and ρ is the density. The radiusof gyration of the beam cross section is κ and is given as t/√{squareroot over (12)}, where t is the thickness of the beam with rectangularcross section. The value of η for the beam is: 1.194²; 2.988²; 5²; 7²; .. . ; (2n−1)², approaching whole numbers for higher η values. Theovertones are not harmonics of the fundamental, and the numerical termsfor f₁ and f₂, which result from the trigonometric solutions involved inthe derivation, must be used without rounding. For the longitudinalmodes of fixed-free beam, f_(n)=(2n−1)c/4L.

From the above equation, and knowing ρ=1.802 g·cm⁻³, we obtained Young'smodulus to be 2 GPa, with the speed of sound, therefore, being 1,100m·s⁻¹; we estimate a 12% uncertainty in Y due to errors in t, L, and f.This value of Young's modulus (N·m⁻²) is very similar to that measuredfor TTF-TCNQ single crystals using a mm-length vibrating reed under analternating voltage. Both materials are pseudo-one-dimensional, and thevalue of the modulus is indicative of the elastic nature along thestacking axis in the direction of weak intercolumn interactions. Young'smodulus slowly varies in value in the temperature range of 50 to 300 Kbut, when extrapolated to higher temperatures, decreases for bothTTF-TCNQ and K(TCNQ). From the absorbed laser pulse energy (30 nJ), theamount of material (7.2×10⁻¹⁴ kg), and assuming the heat capacity to besimilar to TTF-TCNQ (430 J·K⁻¹·mol⁻¹), the temperature rise in themicroscale crystal is expected to be at most 260 K. Finally, we notethat for the same modulus reported here, the frequency of longitudinalmode expansion [f=c/4L; n=1] should be nearly 25 MHz, which is not seenin the FFT with the reported resolution, thus suggesting that theobserved frequencies in the longitudinal direction are those due tocantilever motion in the z direction; the longitudinal expansion of thecrystal is about 1 to 2% of its length, which in this case will be 100to 200 nm.

The potential energy stored in the crystal and the force exerted by thecrystal at the moment of full extension along the long axis just aftertime zero [cf. FIG. 30(A)] can be estimated from the amplitudes andusing Hooke's law:

$\begin{matrix}{V = {\frac{1}{2}\left( \frac{YA}{L} \right)\Delta\; L^{2}}} & \left( {20\; a} \right) \\{F = {\left( \frac{YA}{L} \right)\Delta\; L}} & \left( {20\; b} \right)\end{matrix}$where V and F are the potential energy and force, respectively, and A isthe cross-sectional area of the crystal. The bracketed term in equation(20) is the spring constant (assuming harmonic elasticity, and not theplasticity range), and by simple substitution of the values, we obtained200 pJ and 600 μN for the potential energy and force, respectively,considering the maximum possible expansion of 720 nm; even when theamplitude is at its half value [see FIG. 30(A)], the force is very large(˜300 μN). For comparison, the average force produced by a single myosinmolecule acting on an actin filament, which was anchored by twopolystyrene beads, was measured to be a few piconewtons. In other words,because of molecular stacking, the force is huge. Also because of themicroscale cross-section, the pressure of expansion translates to 0.1GPa, only a few orders of magnitude less than pressures exerted by adiamond anvil. Based on the laser fluence, crystal dimensions, andabsorptivity of Cu(TCNQ) at 671 nm (3.5×10⁶ m⁻¹), the maximum pulseenergy absorbed by the crystal is 30 nJ. This means that, of the initialoptical energy, a minimum of ˜1% is converted into mechanical motion ofthe crystal. But in fact, it could reach 10 or more percent asdetermined by the projection of the electric field of light on thecrystal.

In order to verify the trend in frequency shifts, the above studies wereextended to another set of crystal beams, namely those of reduceddimensions. Because the resonant frequencies of a fixed-free beam aredetermined, in part, by the beam dimensions [cf. equation (19)], aCu(TCNQ) crystal of different length than that shown in FIG. 30 shouldchange the oscillation frequencies by the κ/L² dependence. With asmaller cantilever beam we measured the oscillation frequencies for acrystal of 300 nm thickness and 4.6 μm length, using the same laserparameters as for the larger crystals, and found them to be at highervalues (FIG. 31). This is confirmed by the FFTs of the displacementspanning the range 0 to 3.3 μs [FIGS. 31 (C) and (D)]; a strongresonance near 9 MHz with another weaker resonance at 3.6 MHz in thelongitudinal direction [FIG. 31 (C)] is evident. Within a fewmicroseconds, the only observed frequency in the FFT was near 9 MHz.This oscillation persists up to the time scans of 30 μs, at which pointthe amplitude was still roughly 40% of the leveling value near 2 μs. Bytaking this duration (30 μs) to be the decay time (τ) required for theamplitude to fall to 1/e of the original value, the quality factor(Q=πfτ) of the crystal free oscillator becomes near 1,000. However, onlonger time scales, and with less step resolution, the crystal recoversto the initial state in a few milliseconds, and if the mechanical motionpersists, Q would increase by an order of magnitude.

It is clear from the resonance value of the flexural frequency at 9 MHzthat as the beam reduces in size, the frequency increases, as expectedfrom equation (19). However, if we use this frequency to predict Young'smodulus we will obtain a value of 30 GPa, which is an order of magnitudelarger than that for the larger microscale crystal. The discrepancypoints to the real differences in modes structure as we reachnanometer-scale cantilevers. One must consider, among other things, theanchor-point(s) of the crystals, the frictional force with substrate andother crystals, and the curvature of the beam (see movie in supportinginformation). This curvature will cause the crystal to deviate fromideal Euler-Bernoulli beam dynamics, thus shifting resonance frequenciesfrom their expected positions. Interestingly, by using the value of 30GPa for Young's modulus, the minimum conversion efficiency increases bya factor of 15. These dependencies and the extent of displacement indifferent directions, together with the physics of modes coupling(dephasing and rephasing), will be the subject of our full account ofthis work.

Thus, with 4D electron microscopy it is possible to visualize in realspace and time the functional nanomechanical motions of cantilevers.From tomographic tilt series of images, the crystalline beam stands onthe substrate as defined by the polar and azimuthal angles. Theresonance oscillations of two beams, micro and nanocantilevers, wereobserved in situ giving Young's elastic modulus, the force, and thepotential energy stored. The systems studied are unique 1D molecularstructures, which provide anisotropic and colossal expansions. Thecantilever motions are fundamentally of two types, longitudinal andtransverse, and have resonance Q factors that make them persist for upto a millisecond. The function is robust, at least for 10⁷ continuouspulse cycles (˜10¹¹ oscillations for the recorded frames), with nodamage or plasticity. With these imaging methods in real-time’ and withother variants, it is now possible to test the various theoreticalmodels involved in MEMS and NEMS.

Electron energy loss spectroscopy (EELS) is a powerful tool in the studyof valency, bonding and structure of solids. Using our 4D electronmicroscope, we have performed ultrafast EELS, taking the time resolutionin the energy-time space into the femtosecond regime, a 10 order ofmagnitude increase, and for a table-top apparatus. It is shown that theenergy-time-amplitude space of graphite is selective to changes,especially in the electron density of the π+σ plasmon of the collectiveoscillation of the four electrons of carbon. Embodiments of the presentinvention related to EELS enable the microscope to be used as ananalytical tool. As electrons pass through the specimen, each type ofmaterial (e.g., gold, copper, or zinc) will have a different electronenergy. Thus, it is possible to “tune” into a particular element andstudy the dynamic behavior of the material itself.

In microscopy, EELS provides rich characteristics of energy bandsdescribing modes of surface atoms, valence- and core-electronexcitations, and interferences due to local structural bonding. Thescope of applications thus spans surface and bulk elemental analysis,chemical characterization and electronic structure of solids. Thestatic, time-integrated, EEL spectra do not provide direct dynamicinformation, and with video-rate scanning in the microscope couldchanges be recorded only with a time resolution of millisecond orlonger. Dedicated time-resolved EELS apparatus, without imaging, haveobtained millisecond resolution, being determined primarily by detectorresponse and electron counts. However, for studies of dynamics ofelectronic structure, valency and bonding, the time resolution mustincrease by at least nine orders of magnitude.

We have performed femtosecond resolved EELS (FEELS) using our ultrafastelectron microscope (UEM), developed for 4D imaging of structures andmorphology. Embodiments of the present invention are conceptuallydifferent from time-resolved EELS (termed TREELS) as the time resolutionin FEELS is not limited by detector response and sweep rate. Moreover,both real-space images and energy spectra can be recorded in situ in UEMand with energy filtering the temporal resolution can be made optimum.We demonstrate the method in the study of graphite which displayschanges on the femtosecond (fs) time scale with the delay steps being250 fs. Near the photon energy of 2.4 eV (away from the zero energy losspeak), and similarly for the π+σ plasmon band, the change is observed,but it is not as significant for the π plasmon band. Thus it is possibleto chart the change from zero to thousands of eV and in 3D plots oftime, energy and amplitude; the decrease in EELS intensity at higherenergies becomes the limiting factor. This table-top approach usingelectrons is discussed in relation to recent achievements using soft andhard (optical) X-rays in laboratory and large-scale facilities ofsynchrotrons and free electron lasers.

According to embodiments of the present invention, the probing electronsand the initiating light pulses are generated by a fs laser, and the EELspectra of the transmitted electrons are recorded in a stroboscopic modeby adjusting the time delay between the pump photons and the probeelectron bunches. The concept of single-electron packet used before inimaging is utilized in this approach. When each ultrafast electronpacket contains at most one electron, “the single-electron mode,”space-charge broadening of the zero-loss energy peak, which decreasesthe spectrometer's resolution, is absent.

FIG. 32 is a schematic diagram of a microscope used in embodiments ofthe present invention. A train of 220 fs laser pulses at 1.2 eV wasfrequency doubled and tripled and then split into two beams. In otherembodiments, a range of laser pulse widths could be used, for examplefrom about 10 femtoseconds to about 10 microseconds. The frequencytripled light at 3.6 eV was directed to the microscope photocathode, andthe photoelectron probe pulse was accelerated to 200 keV. The 2.4 eVpulses were steered to the specimen, and provided the excitation at afluence of 5.3 mJ/cm². In other embodiments, a fluence ranging fromabout 1 mJ/cm² to about 20 mJ/cm² could be utilized. By varying thedelay time between the electron and optical pulses, the time dependenceof the associated EEL spectrum was followed. The electrons pass throughthe sample and a set of magnetic lenses to illuminate the CCD camera,forming either a high resolution image of the specimen, a diffractionimage, or they can be energy dispersed to provide the EEL spectra.

The apparatus is equipped with a Gatan imaging filter (GIF) Tridiem, ofthe postcolumn type, which is attached below the transmission microscopecamera chamber. The energy width of near 1 eV was measured for the EELSzero-loss peak and it is comparable to that obtained in thermal-modeoperation of the TEM, but increases significantly in the space-chargelimited regime. The experiments were performed at repetition rates of100 kHz and 1 MHz, and no difference in the EEL spectra or the temporalbehavior was observed, signifying a complete recovery of electronicstructure changes between subsequent pulses. The reported temporalchanges were missed when the scan resolution exceeded 250 fs, and theentire profile of the transient is complete in 2 ps. The electron beampasses through the graphite sample perpendicular to the sample surfacewhile the laser light polarization was parallel to the graphene layers.Finally, the zero of time was determined to the precision of thereported steps, and was observed to track the voltage change in the FEGmodule of the microscope.

The semi-metal graphite is a layered structure, which was prepared asfree-standing film. The thickness of the graphite film was estimatedfrom the EEL spectrum to be 106 nm (inelastic mean-free path of ˜150nm), and the crystallinity of the specimen was verified by observing thediffraction pattern which was indexed as reported. FIG. 33 shows astatic EEL spectrum of graphite taken in UEM. The distinct features areobserved in the spectrum and indeed are typical of the electronicstructure bands of graphite; the in-plane π plasmon is found near 7 eV,while at higher energy, the peak at 27 eV is observed with a shoulder at15 eV. These latter peaks correspond to the π+σ oscillation of the bulkand surface plasmons, respectively. The results are in agreement withthose of literature reports. The bands displayed in different colors(FIG. 33) are the simulations of the profiles with peak positionsreproducing the theoretical values near 7, 15 and 27 eV.

The 3D FEELS map of the time-energy evolution of the amplitude of theplasmon portion of the spectrum (up to 35 eV) is shown in FIG. 34,together with the EEL spectrum taken at negative time. The spectra weretaken at 1 MHz repetition-rate, for a pump fluence of 5.3 mJ/cm² at roomtemperature and for ts=250 fs for each difference frame. The mapreflects the difference for all energies and as a function of time, madeby subtracting a reference EEL spectrum at negative time from subsequentones. The relatively strong enhancement of the energy loss in the lowenergy (electron-hole carriers) region is visible and the change is nearthe energy of the laser excitation. This feature represents the energyloss enhancement due to the creation of carriers by the fs laserexcitation in the ππ* band structure, as discussed below. At higherenergy, the 7 eVπ plasmon peak remains nearly unperturbed by theexcitation, and no new features are observed at the correspondingenergies. For the 27 eVπ+σ bulk plasmon an increased spectral weight atpositive time is visible as a peak in the time-resolved spectrum.

In order to obtain details of the temporal evolution of the differentspectroscopic energy bands, we divided the spectrum into three regions:the low energy region between 2 and 5 eV, the π plasmon region between 6and 8 eV, and the π+σ plasmon region between 20 and 30 eV. The 3D dataare integrated in energy within the specified regions of the spectrum,and the temporal evolution of the different loss features are obtained;see FIG. 35. For regions where changes occur, the time scales involvedin the rise and subsequent decay are similar. In FEELS, the shortestdecay is 700 fs taken with the steps of 250 fs. The duration of theoptical pulse is ˜220 fs, but we generate the UV pulse for electrongeneration through a non-linear response, and it is possible that thepulses involved are asymmetric in shape and that multiphotons are partof the process; full analysis will be made later. We note that theobserved ˜700 fs response indeed reflects the joint response from boththe optical and electron pulses and it is an upper limit for theelectronic change.

It is remarkable that, in FIG. 35, the temporal evolution of theinterlayer spacing of graphite obtained by ultrafast electroncrystallography (UEC) at a similar fluence, i.e. 3.5 mJ/cm², thetimescale of the ultrafast compression corresponds well to the period inwhich the bulk plasmon is out of equilibrium; in this plot the zero oftime is defined by the change of signal amplitude. In graphite, thecharacteristic time for the thermalization of photo-excited electrons isknown to be near 500 fs at low fluences (a few μJ/cm²). When excited byan intense laser pulse, a strong electrostatic force between graphenelayers is induced by the generated electron-hole (carrier) plasma. Thiscauses the structure to be out of equilibrium for nearly 1 ps; astressful structural rearrangement is imposed on the crystal, which, atvery high fluences (above 70 mJ/cm²), has been proposed as a cause ofthe phase transformation into diamond.

Because graphite is a quasi two-dimensional structure, distinct spectralfeatures are visible in EELS. The most prominent and studied peaks arethose at 7 eV and the much stronger one at 27 eV. From the solution ofthe in-plane and out-of-plane components of the dielectric tensor it wasshown, for graphite, that the 7 eV band is a π plasmon, resulting frominterband ππ* transitions in the energy range of 2-5 eV, whereas the 27eV band is a π+σ plasmon dominated by σσ* transitions beyond 10 eV (FIG.5). We note that in this case the plasmon frequencies are not directlygiven by the ππ* and σσ* transition energies as they constitutetensorial quantities. For exampleω _(π+σ) ²= ω _(p) ²+¼(Ω_(π) ²+3Ω_(σ) ²),where ω_(p)=npe²/∈₀m)^(1/2) is the free electron gas plasma frequency;Ω_(π), and Ω_(σ) are the excitation energies for ππ* and σσ*transitions, respectively. For ω_(p), the electron density is np, n isthe number of valence electrons per atom and p is the density of atoms,and ∈₀ is the vacuum dielectric constant. It follows that the density ofoccupied and empty (π, σ, π*, and σ*) states is critical, and that the πPlasmon is from the collective excitation of the π electrons (oneelectron in the p-orbital, with screening corrections) whereas the π+σplasmon is the result of all 4 valence electrons collectively excitedover the coherent length scale of bulk graphite; there are also surfaceplasmons but at different energies.

Recently it was demonstrated, both theoretically and experimentally,that the π and π+σ plasmons are sensitive to the inter-layer separation,but while the former shows some shift of peaks the latter isdramatically reduced in intensity, and, when reaching the graphemelimit, only a relatively small peak at ˜15 eV survives. This isparticularly evident when the momentum transfer is perpendicular to thec-axis, the case at hand and for which the EEL spectrum is very similarto ours. With the above in mind, it is now possible to provide, in apreliminary picture, a connection between the selective fs atomicmotions, which are responsible for the structural dynamics, and changesin the dielectric properties of Plasmon resonances, the electronicstructure.

The temporal behavior, and coherent oscillation (shear modes of ˜1 ps),of c-axis expansion display both contraction and expansion on thepicometer length scale per unit cell. The contraction precedes theexpansion, as shown in FIG. 35, with velocity that depends on thefluence, i.e., the density of carriers. With fs excitation, theelectronic bands are populated anisotropically, and, because of energyand momentum conservation, the carriers transiently excitelarge-momentum phonons, so called strongly coupled phonons. They areformed on the fs time scale (electron-phonon coupling) but decay in ˜7ps. The initial compression suggests that the process is a cooperativemotion and is guided by the out-of-equilibrium structure change dictatedby the potential of excited carriers; in this case ππ* excitation whichweakens c-axis bonding.

The initial atomic compression, when plotted with transient EELS data(FIG. 35), shows that it is nearly in synchrony with the initial change,suggesting that the spacing between layers (c-axis separation) is therate determining step, and that in the first 1 ps, the compressed ‘hardgraphite’ effect is what causes the increase in the amplitude of the π+σplasmon peak. In other words, the decrease of the spectral weight due tothe change of electronic structure upon increasing the interlayerseparation (to form graphene) becomes an increase when the plates arecompressed, because of the enhanced collectiveness of all four valenceelectrons of carbon. The change involves shear motions and it is notsurprising that the π+σ peak (dominated by σσ* excitation) is verysensitive to such changes. The π peak is less influenced as only oneelectron is involved, as discussed above, and the amplitude change isrelatively small. The faster recovery of EEL peaks in 700 fs is,accordingly, the consequence of expansion which ‘decouples’ the π and σsystem. Lastly, the relatively large increase in EEL near the photonenergy is due to carrier excitation (π*) which leads to a loss ofelectron energy at near 3 eV, possibly by electronic excitationinvolving the cy system (FIG. 36). The created carriers cause anincrease in the Drude band as evidenced in the decrease in opticaltransmission.

The demonstration of ultrafast EELS in electron microscopy opens thedoor to experiments that can follow the ultrafast dynamics of theelectronic structure in materials. The fs resolution demonstrates theability of UEM to probe transients on the relevant sub-picosecond timescale, while keeping the energy resolution of EELS. Moreover, theselectivity of change in the collective electron density (for graphite)suggests future experiments, including those with changes inpolarization, shorter optical pulses, core excitation and oxidationsites. We believe that this table-top UEM-EELS should provide themethodology for studies which have traditionally been made usingsynchrotrons (and free electron lasers) especially in the UV and softX-ray regions.

Chemical bonding dynamics are important to the understanding ofproperties and behavior of materials and molecules. Utilizingembodiments of the present invention, we have demonstrated the potentialof time-resolved, femtosecond electron energy loss spectroscopy (EELS)for mapping electronic structural changes in the course o nuclearmotions. For graphite, it is found that changes of milli-electron voltsin the energy range of up to 50 electron volts reveal the compressionand expansion of layers on the subpicometer scale (for surface and bulkatoms). These nonequilibrium structural features are correlated with thedirection of change from sp² [two-dimensional (2D) grapheme] to sp³(3D-diamond) electronic hybridization, and the results are compared withtheoretical charge-density calculations. The reported femtosecond timeresolution of four-dimensional (4D) electron microscopy represents anadvance of 10 orders of magnitude over that of conventional EELS method.

Bonding in molecules and materials is determined by the nature ofelectron density distribution between the atoms. The dynamics involvethe evolution of electron density in space and the motion of nuclei thatoccur on the attosecond and femtosecond time scale, respectively. Suchchanges of the charge distribution with time are responsible for theoutcome of chemical reactivity and for phenomena in the condensed phase,including those of phase transitions and nanoscale quantum effects. Withconvergent-beam electron diffraction, the static pattern ofcharge-density difference maps can be visualized, and using x-rayabsorption and photoemission spectroscopy substantial progress has beenmade in the study of electronic-state dynamics in bulks and on surfaces.Electron energy loss spectroscopy (EELS) is a powerful method in thestudy of electronic structure on the atomic scale, usingaberration-corrected microscopy, and in chemical analysis of selectedsites; the comparison with synchrotron-based near-edge x-ray absorptionspectroscopy is impressive. The time and energy resolutions of ultrafastelectron microscopy (UEM) provide the means for the study of (combined)structural and bonding dynamics.

Here, time-resolved EELS is demonstrated in the mapping of chemicalbonding dynamics, which require nearly 10 orders of magnitude increasein resolution from the detector-limited millisecond response. Byfollowing the evolution of the energy spectra (up to 50 eV) withfemtosecond (fs) resolution, it was possible to resolve in graphite thedynamical changes on a millielectronvolt (subpicometer motion) scale. Inthis way, we examined the influence of surface and bulk atoms motion andobserved correlations with electronic structural changes: contraction,expansion, and recurrences. Because the EEL spectra of a specimen inthis energy range contain information about plasmonic properties ofbonding carriers, their observed changes reveal the collective dynamicsof valence electrons.

Graphite is an ideal test case for investigating the correlation betweenstructural and electronic dynamics. Single-layered grapheme, the firsttwo-dimensional (2D) solid to be isolated and the strongest materialknown, has the orbitals on carbon as sp² hybrids, and in graphite theπ-electron is perpendicular to the molecular plane. Strongly compressedgraphite transforms into diamond, whose charge density pattern is a 3Dnetwork of covalent bonds with sp³ hybrid orbitals. Thus, any structuralperturbation on the ultra-short time scale of the motion will lead tochanges in the chemical bonding and should be observable in UEM.Moreover, surface atoms have unique binding, and they too should bedistinguishable in their influence from bulk atom dynamics.

The experiments were performed on a nm-thick single crystal of naturalhexagonal graphite. The sample was cleaved repeatedly until atransparent film was obtained, and then deposited on a transmissionelectron microscopy (TEM) grid; the thickness was determined from EELSto be 108 nm. The fs-resolved EELS (or FEELS) data were recorded in ourUEM, operating in the single-electron per pulse mode to eliminateBoersch's space charge effect. A train of 220 fs infrared laser pulses(λ=1038 nm) was split into two paths, one was frequency-doubled and usedto excite the specimen with a fluence of 1.5 mJ/cm², and the other wasfrequency-tripled into the UV and directed to the photoemissive cathodeto generate the electron packets. These pulses were accelerated in theTEM column and dispersed after transmission through the sample in orderto provide the energy loss spectrum of the material.

The experimental, static EEL spectra of graphite in our UEM, withgrapheme for comparison, are displayed in FIG. 37A; FIG. 37B shows theresults of theoretical calculations. The spectral feature around 7 eV isthe π Plasmon, the strong peak centered around 26.9 eV is the π+σ bulkplasmon, and the weaker peak on its low energy tail is due to thesurface Plasmon. The agreement between the calculated EEL spectra andthe experimental ones is satisfactory both for graphite and grapheme. Ofrelevance to our studies of dynamics is the simulation of the spectrafor different c-axis separations, ranging from twice as large asnaturally occurring (2c/a; a and c are lattice constraints) to 5 timesas large. This thickness dependence is displayed in FIG. 37B.

As displayed in FIG. 37, the surface and bulk Plasmon bands (between 13and 35 eV) can be analyzed using two Voigt functions, thus defining thecentral position, intensity, and width. At different delay times, wemonitored the changes and found that they occur in the intensity andposition; the width and shape of the two spectral components arerelatively unchanged. FIGS. 37C and 37D, show the temporal changes ofthe intensity for both the surface and bulk plasmons. As noted, thebehavior of bulk dynamics is “out of phase” with that of the surfacedynamics, corresponding to an increase in intensity for the former and adecrease for the latter. Each time point represents a 500-fs change.Within the first 1 ps, the bulk Plasmon gains spectral weight with theincrease in intensity. With time, the intensity is found to return toits original (equilibrium) value. At longer times, a reverse in signoccurs, corresponding to a decrease and then an increase in intensity—anapparent recurrence or echo occurring with dispersion. The intensitychange of the surface plasmon in FIGS. 37C and 37D, shows a πphase-shifted temporal evolution with respect to that of the bulkplasmon.

The time dependence of the energy position of the different spectralbands is displayed in FIG. 38. The least-squares fit converges for avalue of the surface plasmon energy at 14.3 eV and of the bulk plasmonat 26.9 eV. The temporal evolution of the surface plasmon gives no signof energy dispersion, whereas the bulk plasmon is found to undergo firsta blueshift and then a redshift at longer times (FIGS. 38A and 38B). Theoverall energy-time changes in the FEEL spectra are displayed in FIG.39. To make the changes more apparent, the difference between thespectra after the arrival of the initiating laser pulse (time zero) anda reference spectrum taken at −20 ps before time zero is shown. The mostpronounced changes are observed in the region near the energy of thelaser itself (2.39 eV), representing the energy-loss enhancement due tothe creation of carriers by the laser excitation, and in the regiondominated by the surface and bulk plasmons (between 13 and 35 eV).Clearly evident in the 3D plot are the energy dependence as a functionof time, the echoes, and the shift in phase.

A wealth of information has been obtained on the spectroscopy andstructural dynamics of graphite. Of particular relevance here are theresults concerning contraction and expansion of layers probed bydiffraction on the ultra-short time scale. Knowing the amplitude ofcontraction/expansion, which is 0.6 pm at the fluence of 1.5 mJ/cm², andfrom the charge of plasmon energy with interlayer distance (FIG. 37), weobtained the results shown in FIG. 38C. The diffraction data, when nowtranslated into energy change, reproduce the pattern in FIG. 38A, withthe amplitude being within a factor of two. When the layers are fullyseparated, that is, reaching grapheme, the bulk plasmon, as expected, iscompletely suppressed.

The dynamics of chemical bonding can now be pictured. The fs opticalexcitation of graphite generates carriers in the nonequilibrium state.They thermalize by electron-electron and electron-phonon interactions ona time scale found to be less than 1 ps, less than 500 fs, and −200 fs.From our FEELS, we obtained a rise of bulk plasmons in ˜180 fs (FIG.39). The carriers generated induce a strong electrostatic force betweengrapheme layers, and ultrafast interlayer contraction occurs as aconsequence. In FIG. 37D, the increase of the bulk plasmon spectralweight on the fs time scale reflects this structural dynamics ofbond-length shortening because it originates from a denser and more 3Dcharge distribution. After the compression, a sequence of dilatationsand successive expansions along the c axis follows, but, at longer timeslattice thermalization dephases the coherent atomic motions; at a higherfluence, strong interlayer distance variations occur, and graphemesheets can be detached as a result of these interlayer collisions. Thus,the observations reported here reflect the change in electronicstructure: contraction toward diamond and expansion toward grapheme. Theenergy change with time correlates well with the EELS change calculatedfor different interlayer distances (FIG. 37).

We have calculated the charge density distribution for the threerelevant structures. The self-consistent density functional theorycalculations were made using the linear muffin-tin orbitalapproximation, and the results are displayed in FIG. 40. To emphasizethe nature of the changes observed in FEELS, and their connection to thedynamics of chemical bonding, we pictorially display the evolution ofthe charge distribution in a natural graphite crystal, a highlycompressed one, and the extreme case of diamond. Once can see thetransition from a 2D to a 3D electronic structure. The compressed andexpanded graphite can pictorially be visualized to deduce the change inelectron density as interlayer separations change.

With image, energy, and time resolution in 4D UEM, it is possible tovisualize dynamical changes of structure and electronic distribution.Such stroboscopic observations require time and energy resolutions of fsand meV, respectively, as evidenced in the case study (graphite)reported here, and for which the dynamics manifest compression/expansionof atomic planes and electronic sp²/sp³-type hybridization change. Theapplication demonstrates the potential for examining the nature ofcharge density and chemical bonding in the course of physical/chemicalor materials phase change. It would be of interest to extend the scaleof energy from ˜1 eV, with 100 meV resolution, to the hundreds of eV forexploring other dynamical processes of bonding.

The following articles are hereby incorporated by reference for allpurposes:

-   4D imaging of transient structures and morphologies in ultrafast    electron microscopy, Brett Barwick, et al., Science, Vol. 322, Nov.    21, 2008, p. 1227.-   Temporal lenses for attosecond and femtosecond electron pulses,    Shawn A. Hibert, et al., PNAS, Vol. 106, No. 26, Jun. 30, 2009, p.    10558.-   Nanoscale mechanical drumming visualized by 4D electron microscopy,    Oh-Hoon Kwon, et al., Nanoletters, Vol. 8, No. 11, November 2008, p.    3557.-   Nanomechanical motions of cantilevers: direct imaging in real space    and time with 4D electron microscopy, David J. Flannigan, et al.,    Nanoletters, Vol. 9, No. 2 (2009), p. 875.-   EELS femtosecond resolved in 4D ultrafast electron microscopy,    Fabrizio Carbone, et al., Chemical Physics Letters, 468 (2009), p.    107.-   Dynamics of chemical bonding mapped by energy-resolved 4D electron    microscopy, Fabrizio Carbone, et al., Science, Vol. 325, Jul. 10,    2009, p. 181.-   Atomic-scale imaging in real and energy space developed in ultrafast    electron microscopy, Hyun Soon Park, et al., Nanoletters, Vol. 7,    No. 9, September 2007, p. 2545.

It is also understood that the examples and embodiments described hereinare for illustrative purposes only and that various modifications orchanges in light thereof will be suggested to persons skilled in the artand are to be included within the spirit and purview of this applicationand scope of the appended claims.

What is claimed is:
 1. A system for imaging a sample, the systemcomprising: a stage assembly disposed in a chamber and adapted toreceive a sample to be imaged; a first laser source capable of emittinga first optical pulse; a second laser source capable of emitting asecond optical pulse; a cathode operable to emit an electron pulse inresponse to at least one of the first optical pulse or the secondoptical pulse; a lens assembly operable to direct the electron pulse toimpinge on the sample disposed on the stage assembly; and a detectoroperable to capture the electron pulse passing through the sample. 2.The system of claim 1 wherein the system is an electron energy lossspectroscopy (EELS) system.
 3. The system of claim 1 wherein the firstoptical pulse is less than 1 ps in duration and wherein the secondoptical pulse is less than 1 ns in duration.
 4. The system of claim 1wherein the electron pulse is emitted less than 1 ps in duration inresponse to the first optical pulse.
 5. The system of claim 1 whereinthe lens assembly is an electron lens assembly.
 6. The system of claim 1wherein the chamber comprises a vacuum chamber.
 7. The system of claim 1further comprising a processor, the processor being configured toprocess a data signal associated with one or more electrons passingthrough the sample.
 8. The system of claim 7 wherein an image is formedbased in part on the data signal.
 9. A system for imaging a sample, thesystem comprising: a first laser source capable of emitting a firstoptical pulse; a second laser source capable of emitting a secondoptical pulse a cathode capable of emitting an electron pulse inresponse to the first optical pulse or second optical pulse; a lensassembly operable to direct the electron pulse to impinge on the sample;and a detector operable to capture the electron pulse passing throughthe sample.
 10. The system of claim 9 wherein the laser source iscapable of emitting the first optical pulse or second optical pulse ofless than 1 ps in duration.
 11. The system of claim 9 wherein the lasersource is capable of emitting the first optical pulse or second opticalpulse of less than 1 ns in duration.